601 fJ / A / 

sy 1 



2 D io U •■■ 



2si&HE PERSONAL DISTRIBUTION OF 

INCOME IN THE UNITED STATES 



BY 

FREDERICK ROBERTSON MACAULAY 
NATIONAL BUREAU OF ECONOMIC RESEARCH, INC. 



r" 



THE PERSONAL DISTRIBUTION OF 

INCOME IN THE UNITED STATES 



BY 

FREDERICK ROBERTSON MACAULAY 
NATIONAL BUREAU OF ECONOMIC RESEARCH, INC. 



Submitted in Partial Fulfilment of the Requirements for the Degree 
of Doctor of Philosophy in the Faculty of Political 
Science, Columbia University 



m 



NEW YORK 

HARCOURT, BRACE AND COMPANY 

1922 






COPYEIGHT, 1922, BY 
NATIONAL BUREAU OF ECONOMIC RESEARCH, INC. 



^1 ^'s ?M 



Printed in the U. S. A. 



PREFACE 

In the year 1922 the National Bureau of Economic Research, Inc., pub- 
Hshed in two volumes the result of an investigation into "Income in the 
United States." Part III of Volume II of that work consisted of the present 
study. The author acknowledges with thanks the courtesy of the National 
Bureau of Economic Research in permitting him to have this reprint made 
from the original plates.* 

1 This fact explains the pagination. 



TABLE OF CONTENTS 

Chapter Page 

27. The Problem 341 

Practical and theoretical difficulties connected with formulation 

of the problem. Relation of personal distribution to factorial 
distribution, 

28. Pareto's Law and the Problem op Mathematically Describ- 

ing THE Frequency Distribution of Income 344 

Pareto's Law. Improbability that any simple mathema.tical ex- 
pression adequately describing the frequency distribution of in- 
come can ever be formulated. Heterogeneity of the data. 

29. Official Income Censuses 395 

The Australian income census of 1915. 

30. American Income Tax Returns 401 

Pecularities of the tax returns from year to year. Irregularities 

and fluctuations in the distribution of non-reporting and under- 
statement. The under $5,000 and over $5,000 groups. Wages 
and total income, 

31. Income Distributions from Other Sources Than Income Tax 

Returns 415 

Purposes for which existing distributions have been collected make 
them extremely ill adapted to our use — picked data. 

32. Wage Distribution 418 

Relations between rates and earnings, earnings and income. Earn- 
ings per hour, per day, and per week. Distribution of hours 
worked in a week, and weeks worked in a year. Federal returns 

of income by sources. The problem of deriving one regression 
line from the other. 

33. The Construction of a Frequency Curve for All Income 

Recipients 424 

An income census the direct and adequate method of solving the 
problem. Piecing together the existing data. Checking them 
for internal consistency and agreement with collateral informa- 
tion. Conjectural nature of final results. 



CHAPTER 27 
THE PROBLEM 

What is the frequency distribution of annual income among personal 
income recipients in the United States? Before we can give an intelligent 
answer to this question, we must formulate it more definitely by indicating 
certain connotations' which logic or expediency leads us to attach to some 
of its terms. 

By income it seems desirable to mean actual money income, plus the 
estimated money value of the more important of those items of commodity 
or service income on which a money value is ordinarily placed. Two of 
the most important items which are thus included are the annual rental 
values of owned homes and the value of farm produce consumed by farmers' 
families. 

In line with the ordinary convention, we have excluded from our defini- 
tion of income, that income, whether monetary or non-monetary, which a 
wife receives from her husband or a child from its parents.^ Not only is 
such exclusion practically expedient but it is also theoretically defensible 
and that quite apart from the fact that a money value is not ordinarily 
placed on the services of wife or child, wages of housekeepers to the con- 
trary notwithstanding. 

The frequency distribution resulting from the exclusion of such quasi 
incomes will be less heterogeneous and more significant and interpretable 
than the distribution which would result from inclusion. For the relation 
of the incomes of wives and children to the economic struggle is derived 
and secondary, while that of most other incomes is direct and primary. 
Now, though the distribution of income among persons is not synonymous 
with distribution among the factors of production, the two problems are 
very closely related. An individual's income may be thought of as made 
up of wages, rent, interest and dividends, profits, and gifts or allowances. 
If we omit this last type of income, the problem of factorial distribution 
proposes an investigation of how and why the individual received what 
remains. Even if gifts and allowances admitted of any such systematic 
and reasoned explanation as may be given of rent, wages, etc., the ex- 
planation would be of a totally different kind. Hence, for the purposes of 
this investigation, it seems undesirable to classify as income, the receipts, 

1 That is, while such income has, of course, been counted in the first instance as income 
of the husband or parent it has not been re-counted as income of the wife or child. 

341 



342 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

whether monetary or non-monetary, of those persons receiving merely 
allowances or gifts. ^ 

Similar considerations have led us to think of an income recipient as an 
individual rather than a family. Just as it is the husband and not the 
wife, the parent and not the child, so it is the individual and not the family 
who, as an income receiver, comes into direct economic relationship with 
the machinery of distribution. 

The chief argument in favor of family rather than individual treatment 
of the frequency distribution is based upon the idea that, though income 
accrues to the individual and not the family, the family is a more significant 
unit of economic need than the individual. But this is. a different approach 
to the question and has, of course, no intimate relation to the problem of 
factorial distribution. Moreover, we must remember that if we are going 
to improve appreciably upon the individual, even as a need unit, we can- 
not stop with actual biological families with their great variation in size 
and constitution, but must introduce the concept of the theoretical family — 
father, mother and three children, for example. This last concept is, in 
its raw form, quite unusable. The population is not made up of such 
theoretical families. We may discuss what a family of five ought to get 
to maintain a decent standard of living, but we cannot divide the actual 
population into families of five and discuss what these non-existent hy- 
pothetical families actually do get. There remains the alternative of ex- 
pressing actual families in terms of some need unit such as the ''ammain." ^ 
While this last procedure would probably yield an extremely interesting 
distribution based upon need units, it is impractical to attempt any such 
solution with the data available.^ 

Though a distribution of income among actual biological families would 
appear to be somewhat less enlightening and interpretable than a dis- 
tribution by individuals or by ammains, it would have its own peculiar 
interest and we would have attempted its construction had the data been 
adequate for such a purpose. Most of the data bearing on income dis- 
tribution are in the individual form; wages distributions, for example, are 

1 Of course if the wife or child has "independent" income, that income is no longer of the 
nature of a gift or allowance even though it may arise from property originally deeded by 
the husband or father. It is now explainable in terms of rent, interest, etc. 

If income be defined as above, the term personal income recipient will correspond closely 
to the census expression person gainfully ernployed. Perhaps the most important difference 
is that we do not and the Census does include as separate income recipients, farm laborers 
working on the home farm. 

2 Ammain is a word coined by W. I. King and E. Sydenstricker and defined by them, for 
any given class of people, as "a gross demand for articles of consumption having a total 
money value equal to that demanded by the average male in that class at the age when his 
total requirements for expense of maintenance reach a maximum." Measurement of Relative 
Economic Status of Families. Quarterly Publications of the American Statistical Association, 
Sept., 1921, p. 852. 

3 It is of course quite possible to estimate the average per ammain income, as has been done 
by Mr. King; the total income of the people can be divided by the estimated number of 
ammains in the population. See pages 233 and 234. 



THE PROBLEM 343 

almost without exception in that form. Now to estimate the frequency- 
distribution of income among famihes from data which, in the first place, 
are in the individual form and, in the second place, are extremely inade- 
quate for estimating even the distribution among individuals, could only 
increase the degree of uncertainty in our results. 

A few words explaining the reason for introducing the next chapter at 
this point are not out of place here. The data upon which an estimate of 
even the individual distribution of income in the United States must be 
based impress one as being in such shape that it is impossible to arrive at 
more than the roughest sort of approximation by any mere direct adding 
process. Some more ingenious plan would seem almost necessary. For 
example, would it not be possible to formulate a general mathematical 
"law" for the distribution of incomes which law might then be used for 
"adjusting" the tentative and hypothetical results obtained from piecing 
together the existing scanty and inadequate material? 

The possibility and desirability of mathematically describing the fre- 
quency distribution of income would seem intimately tied up with the case 
for mathematically describing error distributions and statistical distribu- 
tions in general. The fact that, in our problem, the "law" would be largely 
derived from the same data as those which were to be "adjusted" need 
not greatly disturb us. The procedure of adjusting observations in the 
light of a mathematical expression derived from the same observations is 
not novel. A number of attempts, one of which has become world-famous, 
have been made to demonstrate that the distribution of income follows 
a definite mathematical law. However, the next chapter will show why 
we fear that no rational and useful mathematical law will soon be formu- 
lated. 



CHAPTER 28 

PARETO'S LAW AND THE GENERAL PROBLEM OF MATHE- 
MATICALLY DESCRIBING THE FREQUENCY DISTRIBU- 
TION OF INCOME 

The problem of formulating a mathematical expression which shall de- 
scribe the frequency distribution of income in all places and at all times, 
not only closely, but also elegantly, and if possible rationally as opposed 
to empirically, has had great attractions for the mathematical economist 
and statistician. The most famous of all attempts at the solution of this 
fascinating problem are those which have been made by Vilfredo Pareto. 
Professor Pareto has been intensely interested in this subject for many 
years and the discussion of it runs through nearly all of his published 
work. The almost inevitable result is that "Pareto's Law" appears in a 
number of slightly different forms and Professor Pareto's feelings con- 
cerning the "law" run all the way from treating it as inevitable and im- 
mutable to speaking of it as "merely empirical." 

In its best known, most famous, and most dogmatic form, Pareto's Law 
runs about as follows: 

L In all countries and at all times the distribution of income is such 
that the upper (income-tax) ranges of the income frequency distribution 
curve may be described as follows: If the logarithms of income sizes be 
charted on a horizontal scale and the logarithms of the numbers of persons 
having an income of a particular size or over be charted on a vertical 
scale, then the resulting observational points will lie approximately along 
a straight line. In other words, if 

X = income size and 

y = number of persons having that income or larger 
then log y = log 6 + m log x 
or y = bx^.'^ 

2. In all countries and at all recent times the slope of this straight line 
fitted to the cumulative distribution, that is, the constant m in the equa- 
tion y = bx^, will be approximately L5.^ 

3. The rigidity and universality of the two preceding conclusions strongly 

1 If the cumulative distribution (cumulating from the higher towards the lower incomes 
as Pareto does) on a double log scale could be exactly described by the equation y = bx"', 
the non-cumulative distribution could be described by the equation Y = — mbx""-^^. 

2 Strictly, minus 1.5, though Pareto neglects the sign. 

344 



PARETO'S LAW 345 

suggest that the shape of the income frequency distribution curve on a 
double log scale is, for all countries and at all times, inevitably the same 
not only in the upper (income-tax) range but throughout its entire length. 

4. If then the nature of the whole income frequency distribution is 
unchanging and unchangeable there is, of course, no possibility of economic 
welfare being increased through any change in the proportion of the total 
income going to the relatively poor. Economic welfare can be increased 
only through increased production. In other words, Pareto's Law in this 
extreme form constitutes a modern substitute for the Wages Fund Doc- 
trine. 

This is the most dogmatic form in which the "law" appears. In his 
later work Professor Pareto drew further and further away from the con- 
fidence of his first position. He had early stated that the straight line did 
not seem adequate to describe distributions from all times and places and 
had proposed more complicated equations.^ He has held more strongly 
to the significance of the similarity of slopes but he has wavered in his 
faith that the lower income portions of the curve (below the income-tax 
minimum) were necessarily similar for all countries and all times. He has 
given up the suggestion that existing distributions are inevitable though 
still speaking of the law as true within certain definite ranges. To translate 
from his Manuel (p. 391): "Some persons would deduce from it a general 
law as to the only way in which the inequality of incomes can be dimin- 
ished. But such a conclusion far transcends anything that can be derived 
from the premises. Empirical laws, like those with which we are here 
concerned, have little or no value outside the limits for which they were 
found experimentally to be true." Indeed Professor Pareto has himself 
drawn attention to so many difficulties inherent in the crude dogmatic 
form of the law that this chapter must not be taken as primarily a criticism 
of his work but rather as a note on the general problem of mathematically 
describing the frequency distribution of incomes. 

Almost as soon as he had formulated his law Professor Pareto recognized 
the impossibility of extrapolating the straight line formula into the lower 
income ranges (outside of the income-tax data which he had been using). 
The straight line formula involves the absurdity of an infinite number of 
individuals having approximately zero incomes. Professor Pareto felt 
that this zero mode with an infinite ordinate was absurd. He believed 
that the curve must have a definite mode at an income size well above 
zero " and with a finite number of income recipients in the modal group. 

1 The inadequacy of these more complicated equations is discussed later. See pp. 348, 363 
and 364. 

2 This is, of course, not absolutely necessary. It depends upon our definitions of income 
and income recipient. If we include the negligible money receipts of young children living 
at home we might possibly have a mode close to zero. There are few children who do not 
really earn a few pennies each year. Compare Chart 31A page 416. 



346 PERSONAL DISTRIBUTION OF INCOME IN U. -S. 

Having come to the conclusion that the income frequency distribution 
curve must inevitably have a definite mode well above zero income and 
tail off in both directions from that mode, Professor Pareto was led to 
think of the possibilities of the simplest of all frequency curves, the normal 
curve of error. However, after examination and consideration, he felt 
strongly that the normal curve of error could not possibly be used. He 
became convinced that the normal curve was not the law of the data for 
the good and sufficient reason that the part of the data curve given by 
income-tax returns is of a radically different shape from any part of a 
normal curve. ^ 

Professor Pareto finds a further argument against using the normal curve 
in the irrationality of such a curve outside the range of the data. 
The mode of the complete frequency curve for income distribution is at 
least as low as the minimum taxable income. Income-tax data prove this. 
However, a normal curve is symmetrical. Hence, if a normal curve could 
describe the upper ranges of the income curve as given by income-tax data 
then in the lower ranges it would cut the ij axis and pass into the second 
quadrant, in other words show a large number of negative incomes. 

Now, aside from the fact that this whole argument is unnecessary if 
the data themselves cannot be described even approximately by a normal 
curve, Professor Pareto's discussion reveals a curious change in his middle 
term. If he had said that a symmetrical curve on a natural scale with a 
mode at least as low as the income-tax minimum would show unbelievably 
large negative incomes we could follow him but when he states that not 
only can there be no zero incomes but that there can be no incomes below 
"the minimum of existence" we realize that he has unconsciously changed 
the meaning of his middle term. Having examined a mass of income-tax 
data, all of which were concerned with net money income and from these 
data having formulated a law, he now apparently without realizing it, 
changes the meaning of the word income from net money income to money 
value of commodities consumed, and assumes that those who receive a money 
income less than a certain minimum must inevitably die of starvation. 

' Though Pareto seems to have thoroughly understood this fact, his discussion is not al- 
together satisfactory. He states that the data for the higher incomes show a larger number 
of such incomes than the normal curve would indicate. This is hardly adequate. To have 
stated that the upper and lower ranges showed too many incomes as compared with the middle 
range would have been better. An easy way to realize clearly the impossibility of describing 
income-tax data by a normal curve is to plot a portion of the non-cumulative data on a natural 
X log y basis. When so charted the data present a concave shaped curve. However, if the 
data were describable by any part of a normal curve of error, they would show a convex ap- 
pearance, or in the limiting case a straight line, as the equation of the normal curve of error 
■ — x"^ 

\2/x = 2/oe """ / becomes, on a natural x log y scale, logej/x = logei/o — 2~2 ^^ ^ second degree 

parabola whose axis is perpendicular to the x axis of coordinates.^ 

The reader must note that the limiting straight line case mentioned above is on a natural 
X log y scale and not (as the Pareto straight line) on a log x log y scale. (Note concluded 
page 347.) 



PARETO'S LAW 



347 



Children receive in general negligible money incomes. Many other persons 
in the community are in the same position. A business man may ''lose 
money" in a given year, in other words he may have a negative money 
income. There seems no essential absurdity in assuming that a large 
number of persons receive money incomes much less than necessary to 

(Note 1 page 346 concluded.) 

Chart 28A showing curves fitted to observations on the heights of men illustrates the ap- 
pearance of the normal curve on a natural scale and on a natural x log y scale. That chart 
also illustrates another fact of importance in this discussion, namely, that fitting to a different 
function of the variable gives a different fit. 



-/so 
~/zo 

-90 



DISTRIBUTION OF HEIGHTS OF 1078 MEN. 

BIOMET/^/m VOm P4/5 (r/JTH£/?S) 

1-NoRMAL Curve ffiTED to Natural Scale 
Data by Method of Momemts.also 
Logs of Same. 

2-Second Degree Parabola Fitted to 
Natural x Log Y Data by Method of 
Least Squares, ALSO Antilogs of5ame. 



CHART 28 A 





SSS / S3S 6as 61.5 6ZS 6iS 64.S 6SS 66.S 675 68S 69.S 70.S 7I.S li.S 73.5 74iS 7$S 7i.S 

HE/GHT //i /NCHES 




58.5 sas eas ets ezs 63S 6*5 



65.S 66.5 67.5 685 695 

H£/GHT W lliCN£5 



T05 lis 72.S 



348 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

support existence. When in 1915 Australia took a census of the incomes 
of all persons "possessed of property, or in receipt of income," over 14 
per cent of the returns showed incomes ''deficit and nil." ^ 

Professor Pareto's realization of the impossibility of describing income 
distributions by means of normal curves led him to the curious conclusion 
that such distributions were somehow unique and could not be explained 
upon any "chance" hypothesis. "The shape of the curve which is fur- 
nished us by statistics, does not correspond at all to the curve of errors, 
that is to say - to the form which the curve would have if the acquisition 
and conservation of wealth depended only on chance." ^ Moreover, while 
Professor Pareto's further suggestion of possible heterogeneity in the data 
corresponds we believe to the facts, his reason for making such a sug- 
gestion, namely that the data cannot be adequately described by a 
normal curve, is irrelevant.'* "Chance" data distributions are no longer 
thought of as necessarily in any way similar to the normal curve. Even 
error distributions commonly depart widely from the normal curve. 
The best known system of mathematical frequency curves, that of 
Karl Pearson, is intended to describe homogeneous material and is 
based upon a probability foundation, yet the normal curve is only 
one of the many and diverse forms yielded by his fundamental 

,. dlogy x+ a 5 
equation = ^ — — — -• 

dx ho-\-l\X-{- 02X- 

While Pareto's Law in its straight line form was at least an interesting 
suggestion, his efforts to amend the law have not been fruitful. His at- 
tempts to substitute log^A^ = log^A — a loge(x -\- a) or even log^A^ = 
logeA — a \oge{x -}- a) — ^x for the simpler log A^ = log A — a log a; 
have not materially advanced the subject.*^ The more complicated curves 
have the same fundamental drawbacks as the simpler one. Among other 
peculiarities they involve the same absurdity of an infinite number of 
persons in the modal interval and none below the mode. Along with the 
doubling of the number of constants, there comes of course the possibility 
of improving the fit within the range of the data. Such improvement is, 
however, purely artificial and empirical and without special significance, 
as can be easily appreciated by noticing the mathematical characteristics 
of the equation. 

A number of other statisticians have at various times fitted different 
types of frequency curves to distributions of income, wages, rents, wealth, 

1 Compare Table 29A. 

2 My italics. 

3 Manuel, p. 385. See also Cours, pp. 416 and 417. 

4 Vid. Cowrs, pp. 416 and 417. .-,.,.. 

6 Professor A. W. Flux in a review of Pareto s Cours a Economie Fohtique (hconomic Journal, 
March, 1897) drew attention to the inadequacy of Pareto's conception of what were and what 
were not "chance" data. 

6 Cf, Cours, vol, II, p. 305, note, 



PARETO'S LAW 349 

or allied data.^ However, no one has advanced such claims for a "law" 
of income " distribution as were at one time made by Professor Pareto. 
When considering the possibility of helpfully describing the distribution 
of income by any simple mathematical expression, one inevitably begins 
by examining "Pareto's Law." It is so outstanding. Let us therefore 
examine Pareto's Law. 

L Do income distributions, when plotted on a double log scale, 
approximate straight lines closely enough to give such approxi- 
mation much significance? 

Before attempting to answer this question it is of course necessary to 
decide how we shall obtain the straight line with which comparisons are 
to be made. 

Professor Pareto fitted straight lines directly by the method of least 
squares to the cumulative distribution plotted on a double log scale. The 
disadvantage of this procedure is that, though one may obtain the straight 
line which best fits the cumulative distribution, such a straight line may be 
anything but an admirable fit to the non-cumulative figures. For example, 
if a straight line be fitted by the method of least squares to Prussian re- 
turns for 1886 (as given by Professor Pareto) the total number of income 
recipients within the range of the data is, according to the fitted straight 
line, only 5,399,000 while the actual number of returns was 5,557,000, 
notwithstanding the fact that Prussia, 1886, is a sample which runs much 
more nearly straight than is usual. How bad the discrepancy may be 
where the data do not even approximate a straight line is seen in Professor 
Pareto's Oldenburg material. There the least-squares straight line fitted 
to the cumulative distribution on a double log scale gives 91,222 persons 
having incomes over 300 marks per annum while the data give only 54,309. 

1 Among others, Karl Pearson, F. Y. Edgeworth, Henry L. Moore, A. L. Bowley, Lucien 
March, J. C. Kapteyn, C. Bresciani, C. Gini, F. Savorgnan. 

2 Professor H. L. Moore, in his Laws of Wages, is concerned primarily with wages not 
income. 

Professor J. C. Kapteyn has presented a pretty but somewhat hypothetical argument sug- 
gesting that the skewness in the income frequency curve should be such that plotting on a 
log X basis would eliminate it. 

"In several cases we feel at once that the effect of the causes of deviation cannot be inde- 
pendent of the dimension of the quantities observed. In such cases we may conclude at once 
that the frequency curve will be a skew one. To take a single example: 

"Suppose 1000 men to begin trading, each with the same capital; in order to see how their 
wealth will be distributed after the lapse of 10 years, consider first what will be their condition 
at some earlier epoch, say at the end of the fifth year. 

"We may admit that a certain trader A will then only possess a capital of £100, while 
another may possess £100,000. 

"Now if a certain cause of gain or loss comes to operate, what will happen? 

"For instance: Let the price of an article in which both A and B have invested their capital, 
rise or fall. Then it will be evident that if the gain or loss of A be £10, that of B will not be 
£10, but £10,000; that is to say, the effect of this cause will not be independent of the capital, 
but proportional to it." 

J. C. Kapteyn, Skew Frequency Curves in Biology and Statistics, p. 13. 



350 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

The reason for this peculiarity of the fit to the cumulative distribution 
becomes clear when we remember that the least-squares straight line may 
easily deviate widely from the first datum point while a straight line giving 
the same number of income recipients as the data must necessarily pass 
through the first datum point. ^ 

A straight line fitted in such a manner that the total number of per- 
sons and total amount of income correspond to the data for these items 
gives what seems a much more intelligible fit. Charts 28B to 28G show 
cumulative United States frequency distributions from the income-tax 
returns for the years 1914 to 1919 on a double log scale (Professor Pareto's 
suggestion). Two straight lines are fitted to each distribution — one a 
solid least-squares line fitted to the cumulative data points and the 
other a dotted line so fitted that the total number of persons and total 
amount of income correspond to the data figures. While the least-squares 
line may appear much the better fit to these cumulative data, a mere 
glance at Tables 28B to 28G will reveal the fact that such a line is, to 
say the least, a less interpretable fit to the non-cumulative distribution.^ 
It is, of course, evident that neither line is in any year a sufficiently good 
fit to the actual non-cumulative distribution to have much significance. 
No mathematics is necessary to demonstrate this.^ 

1 e. g. in the case of Prussia, 1886, the first datum point is a; = "overSOOM" and y = 54,309 
persons. 

2 Professor Warren M. Persons discussed the fit of the least-squares straight line to Professor 
Pareto's Prussian data for 1892 and 1902 in the Quarterly Journal of Economics, May, 1909, 
and demonstrated the badness of fit of that line to those data. 

3 The income returned for the years 1914 and 1915 was estimated from the number of re- 
turns. Income is not given in the reports for those years. 

In fitting straight lines to the data of Tables 28B to 28G the lowest income interval (in 
which married persons making a joint return are exempt) has always been omitted. To have 
included in our calculations these lowest intervals would have increased still further the bad- 
ness of the fit in the other intervals. 



PARETO'S LAW 



351 



CHART 28 B 



■1,000,000 



- 100,000 



■ 10,000 



• 1,000 ? 



■10 



UNITED STATES IMCOME TAX RETURNS 
1914 

CUMULATIVE FREQUENCY DISTRIBUTION 

AMD 

FITTED (Least Squares) STRAIGHT LIME 
Scales Logarithmic. 




3 4 5 



INCOME IN THOUSANDS OF DOLLARS 
10 20 30 40 50 100 200 300 400 500 

■ 'I 1 I . I I - 



1,000 2,000 3,000 



352 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 







CHART 28 C 






UMITED STATES IMCOME TAX RETURNS 






1915 






CUMULATIVE FREQUENCY DISTRIBUTION 






AMD 


-1,000,000 




FITTED Ueast Squares) STRAIGHT LINE 
Scales Logarithmic. 


-100,000 


X 


. 


- 10,000 1 






o 




^v 


NUMBER 




X. 


-lOQ 




x^ 


-10 






2 3 4 5 


10 


INCOME IN THOUSANDS OF DOLLARS 

20 30 40 50 100 200 300 400500 1,000 2,000 



PARETO'S LAW 



353 









CHART 28 D 

UNITED STATES INCOME TAX RETURNS 
ISI6 

CUMULATIVE FREQUENa DISTRIBUTION 
Ano 


-1,000,00» 






FITTED (Least Squares) STRAIGHT LINE 




^^ 




Scales Logarithmic 


-100,000 


^^ 


^v 


V 


1 

RETURNS 








[X. 

o 






^5^ 


- 1.000 w 

z 






X^ 


-100 






^%C 


■10 






INCOME IN THOUSANDS OF DOLLARS 


2 


3 4 5 


«iL-. 


20 30 40 50 100 200 300 4g0 500 1,000 2/|00 3](),004fl,005j000 



354 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



CHART 28 E 



•1.000,000 



-100,000 



- 10,000 



■ 1,000 



-100 



UNITED 5TATE5 IMCOMETAy RETURNS 
1917 

CUMULATIVE fREOUEMCY DISTRIBUTIOM 

AMD 

FITTED (Least Squares) STRAIGHT LIME 
Scales Logarithmic 




INCOME IN THOUSANDS OF DOLLARS 
10 20 30 40 50 100 200 300 400 500 

' ■ ■ ' .... 



1,000 



2,000 3,0004,000 

I II 



PARETO'S LAW 



355 







CHART 28 F 






UMITED STATES INCOME TAX RETURMS 






ISIS 






CUMULATIVE FREQUEMCY DISTRIBUTIOM 






AND 


1,000,000 >v 


^ 


FITTED (Least Squares) STRAIGHT LINE 
Scales Lo^arHhmic. 


100,000 


\ 


V 


o 

RETURNS 






NUMBER OF 




\, 


-100 




\k 


-10 




■ X 


2 3 4 5 


10 


INCOME IN THOUSANDS OF DOLLARS 
20 30 40 50 100 200 300 400 500 1,000 2,000 3,000 



856 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 







CHART 28 G 






UNITED STATES II1C0ME TAX RETURNS 
1913 


- 1,000,000 ^^ 


V 


CUMULATIVE FREQUEMCY DISTRIBUTION 

AND 

FITTED (Least Squares) STRAIGHT LINE 
Scales Logarithmic. 


- 100,000 


A 


\ 


NUMBER OF RETTURNS 






-100 




%^ 


-10 




^ 


2 3 4 5 


10 


INCOME IN THOUSANDS OF DOLLARS 

20 30 40 50 100 200 300 400500 LOOO 2,000 3,000 



PARETO'S LAW 



357 



TABLE 28B 



UNITED STATES INCOME-TAX RETURNS, 1914 





A 


B 


C 






Income class 


U. S. in- 
come-tax 
returns 


Least -squares 
straight line 


Straight line 

giving 

correct total 

returns and 

income 


Per 

cent 
A is 
of B 


Per 

cent 
A is 
of C 


$ 3,000-$ 4,000 

4,000- 5,000 

5,000- 10,000 

10,000- 15,000 

15,000- 20,000 

20,000- 25,000 

25,000- 30,000 

30,000- 40,000 

40,000- 50,000 

50,000- 100,000 

100,000- 150,000 

150,000- 200,000 

200,000- 250,000 

250,000- 300,000 

300,000- 400,000 

400,000- 500,000 

500,000-1,000,000 

1,000,000 and over 


(82,754) 

66,525 

127,448 

34,141 

15,790 

8,672 

5,483 

6,008 

3,185 

5,161 

1,189 

408 

233 

130 

147 

69 

114 

60 


101,241 

160,545 
38,630 
15,853 
8,230 
4,879 
5,380 
2,793 
4,430 
1,085,5 
437.3 
227.1 
134.6 
148.46 
77.08 
122.20 
62.78 


84,683 
115,347 
32,716 
14,102 
7,589 
4,631 
5,267 
2,835 
4,756 
1,241 
535 
288.1 
175.5 
199.9 
107.6 
180.4 
107.5 


65.7 

79.4 

88.4 

99.6 

105.4 

112.4 

111.7 

114.0 

116.5 

111.6 

92.8 

102.6 

96.6 

99.0 

89.5 

93.3 

95.6 


78.6 

110.5 

104.4 

112.0 

114.3 

118.4 

114.1 

112.3 

108.5 

95.8 

75.9 

80.9 

74.1 

73.5 

64.1 

63.2 

55.8 


Total (over $4,000) 


274,761 


344,256.00 


274,761.0 







358 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



TABLE 28C 



UNITED STATES INCOME-TAX RETURNS, 1915 





A 


B 


C 






Income class 


U. S. in- 
come-tax 
returns 


Least- 
squares 
straight line 


Straight line 

giving 

correct total 

returns and 

income 


Per 

cent 
A is 
of B 


Per 

cent 
A is 
of C 


$ 3,000-$ 4,000 

4,000- 5,000 

5,000- 10,000 

10,000- 15,000 

15,000- 20,000 

20,000- 25,000 

25,000- 30,000 

30,000- 40,000 

40,000- 50,000 

50,000- 100,000 

100,000- 150,000 

150,000- 200,000 

200,000- 250,000 

250,000- 300,000 

300,000- 400,000 

400,000- 500,000 

500,000-1,000,000 

1,000,000 and over 


(69,045) 

58,949 

120,402 

34,102 

16,475 

9,707 

6,196 

7,005 

4,100 

6,847 

1,793 

724 

386 

216 

254 

122 

209 

120 


92,064 
154,507 
40,358 
17,406 
9,372 
5,716 
6,508 
3,503 
5,880 
1,536 
662.5 
356.6 
217.5 
247.7 
133.3 
223.8 
133.6 


68,540 
119,634 
33,013 
14,724 
8,124 
5,050 
5,875 
3,241 
5,653 
1,550 
695.4 
383.8 
238.6 
277.6 
153.2 
267.1 
177.3 


64.0 

77.9 

84.5 

94.7 

103.6 

108.4 

107.6 

117.0 

116.4 

116.7 

109.3 

108. 2 

99.3 

102.5 

91.5 

93.4 

89.8 


86.0 

100.6 

103.3 

111.9 

119.5 

122.7 

119.2 

126.5 

121.1 

114.9 

104.1 

100.6 

90.5 

91.5 

79.6 

78.2 

67.7 


Total (over $4,000) 


267,607 


338,825.0 


267,607.0 







PARETO'S LAW 



359 



TABLE 28D 



UNITED STATES INCOME-TAX RETURNS, 1916 





A 


B 


C 








U. S. in- 
come-tax 
returns 




Straight line 


Per 


Per 


Income class 


Least-squares 
straight line 


giving correct 
total returns 


cent 
A is 


cent 
A is 






and income 


of B 


of C 


$ 3,000-$ 4,000 


(85,122) 










4,000- 5,000 


72,027 


139,096 


86,588 


51.8 


83,2 


5,000- 6,000 


52,029 


84,759 


54,221 


61.4 


96,0 


6,000- 7,000 


36,470 


56,533 


36,899 


64.5 


98.8 


7,000- 8,000 


26,444 


39,846 


26,516 


66.4 


99,7 


8,000- 9,000 


19,959 


29,292 


19,801 


68.1 


100.8 


9,000- 10,000 


15,651 


22,529 


15,445 


69.5 


101.3 


10,000- 15,000 


45,309 


60,668 


42,879 


74.7 


105.7 


15,000- 20,000 


22,618 


26,120 


19,311 


86.6 


117.1 


20,000- 25,000 


12,953 


14,044 


10,726 


92.2 


120.8 


25,000- 30,000 


8,055 


8,558 


6,705 


94.1 


120.1 


30,000- 40,000 


10,058 


9,731 


7,854 


103.5 


128.2 


40,000- 50,000 


5,611 


5,232 


4,362 


107.2 


128.6 


50,000- 60,000 


3,621 


3,189 


2,730 


113,5 


132.6 


60,000- 70,000 


2,548 


2,126 


1,857 


119.8 


137.2 


70,000- 80,000 


1,787 


1,499 


1,334.8 


119.2 


133.9 


80,000- 90,000 


1,422 


1,102 


996.8 


129.0 


142.7 


90,000- 100,000 


1,074 


847 


777.5 


126,8 


138.1 


100,000- 150,000 


2,900 


2 282 1 


2,158.4 


127.1 


134.4 


150,000- 200,000 


1,284 


'982^6 


972.1 


130.7 


132.1 


200,000- 250,000 


726 


528.2 


539.9 


137.4 


134.5 


250,000- 300,000 


427 


321.9 


337.6 


132.6 


126.5 


300,000- 400,000 


469 


366.1 


395.3 


12^.1 


118.6 


400,000- 500,000 


245 


193.8 


219.6 


124.5 


111.6 


500,000-1,000,000 


376 


329.6 


387.4 


114.1 


97.1 


1,000,000-1,500,000 


97 


85.83 


108.7 


113.0 


89.2 


1,500,000-2,000,000 


42 


36.96 


48.88 


113,6 


85.9 


2,000,000-3,000,000 


34 


31.98 


44.19 


106.3 


76.9 


3,000,000-4,000,000 


14 


13.77 


19.91 


101,7 


70.3 


4,000,000-5,000,000 


9 


7.40 


11.05 


121.6 


81.4 


5,000,000 and over 


10 


19.76 


32.87 


50.6 


30.4 


Total (over $4,000) 


344,279 


510,374.00 


344,279.00 







360 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



TABLE 28E 



UNITED STATES INCOME-TAX RETURNS, 1917 





A 


B 


C 








U.S. 

income-tax 

returns 




Straight line 


Per 


Per 


Income class 


Least-squares 
straight line 


giving correct 
total returns 


cent 
A is 


cent 
A is 






and income 


of B 


of C 


$ 1,000-$ 2,000 


(1,640,758) 










2,000- 2,500 


480,486 


618,069 


517,512 


77.7 


92.8 


2,500- 3,000 


358,221 


367,835 


284,620 


97.4 


125.9 


3,000- 4,000 


374,958 


407,366 


376,117 


92.0 


99.7 


4,000- 5,000 


185,805 


212,569 


184,854 


87.4 


100.5 


5,000- 6,000 


105,988 


126,507 


111,097 


83.8 


95.4 


6,000- 7,000 


64,010 


82,746 


73,355 


77.4 


87.3 


7,000- 8,000 


44,363 


57,357 


51,285 


77.3 


86.5 


8,000- 9,000 


31,769 


41,556 


37,362 


76.4 


85.0 


9,000- 10,000 


24,536 


31,551 


28,551 


77.8 


85.9 


10,000- 11,000 


19,221 


24,097 


21,900 


79.8 


87.8 


11,000- 12,000 


15,035 


19,412 


17,747 


77.5 


84.7 


12,000- 13,000 


12,328 


15,707 


14,440 


78.5 


85.4 


13,000- 14,000 


10,427 


12,751 


11,761 


81. S 


88.7 


14,000- 15,000 


8,789 


10,709 


9,909 


82.1 


88.7 


15,000- 20,000 


29,896 


34,161 


31,891 


87.5 


93.7 


20,000- 25,000 


16,806 


17,825 


16,876 


94.3 


99.6 


25,000- 30,000 


10,.571 


10,609 


10,159 


99.6 


104.1 


30,000- 40,000 


12,733 


11,749 


11,385 


108.4 


111.8 


40,000- 50,000 


7,087 


6,130 


6,021 


115.6 


117.7 


50,000- 60,000 


4,541 


3,649 


3,622 


124.4 


125.4 


60,000- 70,000 


2,954 


2,387 


2,391 


123.8 


123.5 


70,000- 80,000 


2 222 


1,653.5 


1,672 


134.4 


132.9 


80,000- 90,000 


l'539 


1,198.5 


1,217.9 


128.4 


126.4 


90,000- 100,000 


1,183 


910.0 


930.8 


130.0 


127.1 


100,000- 150,000 


3,302 


2,384.4 


2,469.5 


138.5 


133.7 


150,000- 200,000 


1,302 


985.2 


1,0.39.6 


132.2 


125.2 


200,000- 250,000 


703 


514.1 


550.5 


136.7 


127.7 


250,000- 300,000 


342 


305.9 


330.8 


111.8 


103.4 


300,000- 400,000 


380 


338.9 


371.2 


112.1 


102.4 


400,000- 500,000 


179 


176.8 


196.3 


101.2 


91.2 


500,000- 750,000 


225 


199.96 


225.56 


112.5 


99.8 


750,000-1,000,000 


90 


82.61 


94.97 


108.9 


94.8 


1,000,000-1,500,000 


67 


68.77 


80.51 


97.4 


83.2 


1,500,000-2,000,000 


33 


28.42 


33.90 


116.1 


97.3 


2,000,000-3,000,000 


24 


23.65 


28.71 


101.5 


83.6 


3,000,000-4,000,000 


5 


9.77 


12.10 


51.2 


41.3 


4,000,000-5,000,000 


8 


5.10 


6.40 


156.9 


125.0 


5,000,000 and over 


4 


12.42 


16.25 


32.2 


24.6 


Total (over $2,000) 


1,832,132 


2,123,640.00 


1,832,132.00 







PARETO^S LAW 



331 



TABLE 28F 



UNITED STATES 


INCOME-TAX RETURNS, 


1918 






A 


B 


C 








U. S. 




Straight line 


Per 


Per 


Income class 


income-tax 


Least-squares 


giving correct 


cent 


cent 




returns 


straight line 


total returns 
and income 


A is 
of B 


A is 
of C 


$ 1,000-$ 2,000 


(1,516,938) 










2,000- 3,000 


1,496,878 


1,375,372 


1,470,366 


108.8 


101.8 


3,000- 4,000 


610.095 


537,892 


566,044 


113.4 


107.8 


4,000- 5,000 


322,241 


269,674 


280,477 


119.5 


114.9 


5,000- 6,000 


126,554 


155,513 


160,366 


81.4 


78.9 


6,000- 7,000 


79,152 


99,102 


101,389 


79.9 


78.1 


7,000- 8,000 


51,381 


67,184 


68,258 


76.5 


75.3 


8,000- 9,000 


35,117 


47,740 


48,266 


73.6 


72.8 


9,000- 10,000 


27,152 


35,628 


35,795 


76.2 


75.9 


10,000- 11,000 


20,414 


28,793 


26,832 


76.2 


76.1 


11,000- 12,000 


16,371 


21,283 


21,231 


76.9 


77.1 


12,000- 13,000 


13,202 


16,999 


16,873 


77.7 


78.2 


13,000- 14,000 


10,882 


13,638 


13,515 


79.8 


80.5 


14,000- 15,000 


9,123 


11,328 


11,165 


80.5 


81.7 


15,000- 20,000 


30,227 


35,214 


34,486 


85.8 


87.7 


20,000- 25,000 


16,350 


17,654 


17,097 


92.6 


95.6 


25,000- 30,000 


10,206 


10,181 


9,762 


100.2 


104.5 


30,000- 40,000 


11,887 


10,886 


10,336 


109.2 


115.0 


40,000- 50,000 


6,449 


5,458 


5,121 


118.2 


125.9 


50,000- 60,000 


3,720 


3,147 


2,928 


118.2 


127.0 


60,000- 70,000 


2,441 


2,006 


1,852 


121.7 


131.8 


70,000- 80,000 


1,691 


1,359.5 


1,246 


124.4 


135.7 


80,000- 90,000 


1,210 


966.2 


881.4 


125.2 


137.3 


90,000- 100,000 


934 


721.0 


653.7 


129.5 


142.9 


100,000- 150,000 


2,358 


1,822.3 


1,636.3 


129.4 


144.1 


150,000- 200,000 


866 


712.7 


629.8 


121.5 


137.5 


200,000- 250,000 


401 


357.3 


312.1 


112.2 


128.5 


250,000- 300,000 


247 


208.0 


178.3 


119.9 


138.5 


300,000- 400,000 


260 


220.3 


188.7 


118.0 


137.8 


400,000- 500,000 


122 


110.5 


93.55 


110.4 


130.4 


500,000- 750,000 


132 


119.28 


99.70 


110.7 


132.4 


750,000-1,000,000 


46 


46.66 


38.36 


98. 6 


119.9 


1,000,000-1,500,000 


33 


36.88 


29.88 


89.5 


110.4 


1,500,000-2,000,000 


16 


14.42 


11.50 


111.0 


139.1 


2,000,000-3,000,000 


11 


11.40 


8.96 


96.5 


122.8 


3,000,000-4,000,000 


4 


4.46 


3.44 


89.7 


116.3 


4,000,000-5,000,000 


2 


2.24 


1.71 


89.3 


117.0 


5,000,000 and over 


1 


4.86 


3.60 


20.6 


27.8 


Total (over $2,000) 


2,908,176 


2,769,408.00 


2,908,176.00 







362 



PERSONAL DISTRlBtJTrON OF INCOME IN U. S. 



TABLE 28G 



UNITED STATES INCOME-TAX RETURNS, 1919 





A 


B 


C 








U.S. 
income-tax 




Straight line 


Per 


Per 


Income class 


Least-squares 
straight line 


giving correct 
total returns 


cent 
A is 


cent 
A Is 




returns 




and income 


of B 


of C 


$ 1,000-$ 2,000 


(1,924,872) 










2,000- 3,000 


1,569,741 


1,984,285 


1,673,688 


79.1 


93.8 


3,000- 4,000 


742,334 


764,739 


660,950 


97.1 


112.3 


4,000- 5,000 


438,154 


379,330 


333,645 


115.5 


131.3 


5,000- 6,000 


167,005 


216,921 


193,470 


77.0 


86.3 


6,000- 7,000 


109.674 


137.278 


123,953 


79.9 


88.5 


7,000- 8,000 


73,719 


92;511 


84,273 


79.7 


87.5 


8,000- 9,000 


50,486 


65,403 


60,066 


77.2 


84.1 


9,000- 10,000 


37,967 


48,583 


44,980 


78.1 


84.4 


10,000- 11,000 


28,499 


36,386 


33,887 


78.3 


84.1 


11,000- 12,000 


22,841 


28,796 


27,027 


79.3 


84.5 


12,000- 13,000 


18,423 


22,921 


21,600 


80.4 


85.3 


13,000- 14,000 


15,248 


18,329 


17,395 


83.2 


87.7 


14,000- 15,000 


12,841 


15,181 


14,459 


84.6 


88.8 


15,000- 20,000 


42,028 


46,868 


45,162 


89.7 


93.1 


20,000- 25,000 


22,605 


23,249 


22,797 


97.2 


99.2 


25,000- 30,000 


13,769 


13,294 


13,228 


103.6 


104.1 


30,000- 40,000 


15,410 


14,084 


14,219 


109.4 


108.4 


40,000- 50,000 


8,298 


6,986 


7,178 


118.8 


115.6 


50,000- 60,000 


5,213 


3,994 


4,162 


130.5 


125.3 


60,000- 70,000 


3,196 


2,528 


2,665 


126.4 


119.9 


70,000- 80,000 


2,237 


1,704 


1,813 


131.3 


123.4 


80,000- 90,000 


1,561 


1,205 


1,292 


129.5 


120.8 


90,000- 100,000 


1,113 


894 


968.3 


124.5 


114.9 


100,000- 150,000 


2,983 


2,240 


2,461.5 


133.2 


121.2 


150,000- 200,000 


1,092 


863.2 


971.6 


126.5 


112.4 


200,000- 250,000 


522 


428.1 


490.4 


121.9 


106.4 


250,000- 300,000 


250 


245.0 


284.4 


102.0 


87.9 


300,000- 400,000 


285 


259.2 


306.0 


110.0 


93.1 


400,000- 500,000 


140 


128.6 


154.4 


108.9 


90.7 


500,000- 750,000 


129 


137.32 


168.2 


93.9 


76.7 


750,000-1,000,000 


60 


52.89 


66.4 


113.4 


90.4 


1,000,000-1,500,000 


34 


41.25 


52.95 


82.4 


64.2 


1,500,000-2,000,000 


13 


15.89 


20.90 


81.8 


62.2 


2,000,000-3,000,000 


7 


12.40 


16.68 


56.5 


42.0 


3,000,000 and over 


11 


12.15 


17.27 


90.5 


63.7 


Total (over $2,000) 


3,407,888 


3,929,905.00 


3,407,888.00 







PARETO'S LAW 



363 



Why do the least-squares straight Knes appear graphically such good 
fits to the cumulative distributions (for at least the later years) when a 
merely arithmetic analysis shows even this fit to the cumulative data to 
be so illusory? Because the percentage range in the number of persons is so 
extremely wide. The deviations of the cumulative data on a double log 
scale from the least-squares straight line are minute when compared with 
the percentage changes in the data from the smallest to the largest incomes. 
But this is not helpful. The fact that there are 100,000 times as many 
persons having incomes over $2,000 per annum as there are persons 
having incomes over $5,000,000 per annum, does not make a theoretical 
reading for a particular income interval of twenty or thirty per cent over 
or under the data reading an unimportant deviation. Charting data on 
a double log scale may thus become a fertile source of error unless ac- 
companied by careful interpretation.^ This fact has long been recognized 
by engineers and others who have had much experience with similar prob- 
lems in curve fitting. 

Another matter of some importance must be noted here. The devia- 
tions of the data from the straight lines might be much less than they are 
and yet constitute extremely bad fits. The data points {even on a non- 
cumulative basis) do not flutter erratically from side to side of the fitted lines; 
they run smoothly, passing through the fitted line at small angles in the way 
that one curve cuts another. Now, in curve fitting, such a condition always 
strongly suggests that the particular mathematical curve used is not in 
any sense the "law" of the data. 

2. Are the slopes of the straight lines fitted to income data 
from different times and places similar in any significant degree? 

1 The dangers of fitting curves with such a combination as a cumulative distribution and 
a double log scale, without further analysis, is well illustrated by the results Professor Pareto 
obtained for Oldenburg. To the Oldenburg data he fitted the rather complicated equation 
log N = log A — a log (x + a) — 8x and obtained the following results. (The value Pareto 
gives for /3, namely .0000631, does not check with his calculated figures given below. = 
.0000274 is evidently what he intended.) 





N 


Logarithms of N 




marks (over) 


Observed 


Calculated 


A 


300 

600 

900 

1,500 

3,000 

6,000 

9,000 

15,300 

30,000 


54,309 

24,043 

16,660 

9,631 

3,502 

994 

445 

140 

25 


4.7349 
4.3810 
4.2217 
3.9837 
3 . 5443 
2.9974 
2.6484 
2.1461 
1.3979 


4 . 7349 
4.4368 
4.2304 
3.9409 
3 . 5008 
2.9997 
2.6671 
2.1838 
1 . 3364 


— .0558 

— .0086 
+ . 0428 
+ .0435 

— .0023 

— .0187 

— .0377 
+ .0615 



(From Cours d' Economie Politique, vol. II, p. 307.) 

The above table may give the reader a vague idea that the fit is rather good. However, 
from the above table the following table may be directly derived : 
TNote concluded page 364.) 



364 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



If income distributions charted on a double log scale not only cannot 
be approximately represented by straight lines, but also differ radically 

(Note 1 page 363 concluded.) 





Number of persons 




Income in marks 


Actual 


Computed 


of computed 


300- 600 

600- 900 

900- 1,500 

1,500- 3,000 

3,000- 6,000 

6,000- 9,000 

9,000-15,300 

15,300-30,000 

Over 30,000 


30,266 

7,383 

7,029 

6,129 

2,508 

549 

305 

115 

25 


26,969 

10,342 

8,270 

5,560 

2,169 

534 

312 

131 

22 


112.2 

71.4 

85.0 

110.2 

115.6 

102.8 

97.8 

87.8 

113.6 


Total 


54,309 


54,309 


100.0 



The fit no longer impresses one as quite so good. See Chart 2SH below. 




PARETO'S LAW 



365 



in shape, it is of course not of great importance whether the straight Unes 
fitted to such data from different times and places have or have not ap- 
proximately constant slopes. For example, a comparison of Chart 28C 
showing the cumulative distribution of United States income-tax returns 
for 1915 on a double log scale and Chart 28F showing similar data for 
1918, makes it plain that, even were the slopes of the fitted straight lines 
for the two years identical, the data curves would still be so different as 
to make the similarity of slope of the fitted lines of almost no significance.^ 
In considering slopes, let us examine further both the data and the 
fitted lines for these two years 1915 and 1918. Tables 281 and 28J give 
some numerical illustrations of the differences between the distributions 
for the two years. Table 281 gives the number of returns in each income 
interval each year and the percentages that the 1918 figures are of the 
1915 figures. 

TABLE 281 

COMPARISON OF UNITED STATES INCOME-TAX RETURNS FOR 

1915 AND 1918 





Number of returns 


Ratio of 1918 




1915 


1918 


to 1915 


$ 4,000 a-$ 5,000 


58,949 

120,402 

.34,102 

16,475 

9,707 

6,196 

7,005 

4,100 

6,847 

1,793 

724 

386 

216 

254 

122 

209 

120 


322,241 

319,356 

69,992 

30,227 

16,350 

10,206 

11,887 

6,449 

9,996 

2,358 

866 

401 

247 

260 

122 

178 

67 


5 4664 


5,000- 10,000 


2 6524 


10,000- 15,000 


2 0524 


15,000- 20,000 


1 8347 


20,000- 25,000 


1 . 6844 


25,000- 30,000 


1 6472 


30,000- 40,000 


1 6969 


40,000- 50,000 


1 5729 


50,000- 100,000 


1 4599 


100,000- 150,000 


1 3151 


150,000- 200,000 


1 1961 


200,000- 250,000 


1 0389 


250,000- 300,000 


1 1435 


300,000- 400,009 


1 0236 


400,000- 500,000 


1 0000 


500,000-1,000,000 


8517 


1,000,000 and over 


.5583 



" The $3,000-$4,000 class is not included, as in 1915 married persons in that class were 
exempted while in 1918 they were not. 



The change as we pass from the $4,000-15,000 interval, where the 1918 
figures are nearly five-and-a-half times the 1915 figures, to the intervals 
above $500,000, where the 1918 figures are actually less than the 1915 
figures, illustrates the great and fundamental difference between the slopes 
of the two distributions. However, such a comparison of unadjusted 

' Compare also the deviations from the fitted lines as given in Tables 28C and 28F. 



366 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



money intervals, while it throws into relief the differences in slope of the 
two distributions, is by no means as enlightening for purposes of exhibiting 
their other essential dissimilarities as a comparison of the two sets of data 
after they have been adjusted for changes in average (per capita) income 
and changes in population. Table 28J gives some comparisons between the 
data for the two years and between the fitted lines for the two years on 
such an adjusted basis. Two intervals, one in the relatively low income 
range and the other in the high income range, are used to illustrate the 
essentially different character of the distributions for the two years. 

TABLE 28J 



COMPARISONS OF UNITED STATES INCOME-TAX RETURNS FOR THE YEARS 1915 AND 
1918 ADJUSTED FOR CHANGES IN AVERAGE (PER CAPITA) INCOME AND CHANGES 
IN POPULATION 



ACTUAL INCOME-TAX DATA 



Income intervals 


Number of returns 
(1) (2) 


Fraction of population 
(3) (4) 


Ratio of 

Column (4) 

to Column (3) 




1915 


1918 


1915 


1918 




Between 12 and 13 
times average income 


21,190 


31,197 


.00021099 


. 00029945 


1.4193 


Between 1,200 and 1,300 
times average income 


43 . 85 


20.37 


.0000004366 


.0000001955 


.4478 


Over 12 times average 
income 


248,600 


271,452 


.00247536 


.00260561 


1 . 0526 




Amount in dollars 


Per cent of total income 




Over 12 times average 
income 


1915 
$4,283,010,735 


1918 
$5,312,832,516 


1915 

11.9% 


1918 

8.7% 


. 7311 



LEAST-SQUARES STRAIGHT LINES 



Income intervals 


Number of returns 
(1) (2) 


Fraction of population 
(3) (4) 


Ratio of 
Column (4) 
to Column (3) 




1915 


1918 


1915 


1918 




Between 12 and 13 
times average income 


32,886 


41,730 


.00032745 


. 00040056 


1.2233 


Between 1,200 and 1,300 
times average income 


47.63 


17.10 


,0000004743 


.0000001041 


.3460 



STRAIGHT LINES FITTED TO GIVE THE SAME TOTAL NUMBER OF RETURNS AND THE 
SAME TOTAL INCOME AS THE INCOME-TAX DATA 



Income intervals 


Number of returns 
(1) (2) 


Fraction of population 
(3) (4) 


Ratio of 

Column (4) 

to Column (3) 




1915 


1918 


1915 


1918 




Between 12 and 13 
times average income 


24,510 


42,460 


.00024405 


.00040756 


1 . 6700 


Between 1,200 and 1 300 
times average income 


54.73 


14.15 


.0000005450 


.0000001358 


.2492 



PARETO'S LAW 



367 





NOTES TO TABLE 28J 
"Average Income" Intervals 






1915 


1918 




$ 358 

4,296 

4,654 

429,600 

465.400 


$ 586 




7,032 


13 " " " 


7,618 


1,200 " " " 


703,200 


1,300 " " " 


761,800 









Equations of Fitted Straight Lines on a Cumulative Double Log Basis 




Least- squares lines 


Lines giving correct total 

number of returns and 

total income 


1914 


y = 11.153.322— 1.559256 x 
y = 10.643299 — 1.419579 X 
y = 10.839435— 1.424638 X 
y = 11.410606— 1.539996 X 
y = 12 . 033697 — 1 . 693823 x 
y = 12 . 320963 — 1 . 734802 x 


y = 10.557242 — 1.420936 x 


1915 


y = 10 . 202382 — 1 . 325598 x 


1916 

1917 


y = 10.212702 — 1.298088 x 
y = 1 1 . 170980 — 1 . 486817 x 


191^. . 


y = 12. 202152 — 1.738497 X 


1919 


y = 12.0361.55 — 1.667258 X 







Table 28J needs little discussion. In the section treating actual income- 
tax data we notice that while the adjusted number of returns in the lower 
income interval ^ increased 41.93 per cent from 1915 to 1918, the adjusted 
number of returns in the upper income interval " decreased 55.22 per cent. 
Moreover, while the adjusted total number of returns above the " 12-times- 
average-income" point increased 5.26 per cent, the adjusted amount of 
income reported in these returns decreased 26.89 per cent. 

Such figures suggest a rather radical change in the distribution of in- 
come during this short three-year period. Similar conclusions may be 
drawn from the figures for the two pairs of fitted lines, though we must 
of course remember that these lines describe only very inadequately the 
actual data. The lines so fitted as to give each year the same total number 
of returns and total amount of income as the data for that year yield 
sensational results. While the adjusted number of returns in the lower 
income-interval increased 67 per cent, the adjusted number of returns 
in the upper income-interval decreased 75.08 per cent. 

Finally, it has been suggested that changes in the characteristics of the 
tax-income-distribution in the United States from 1915 to 1918 may be 
accounted for as the results of the increase in the surtax rates with 1917. 
We do not believe any large part of these changes can be so accounted 
for. Notwithstanding the fact that the country entered the European 
war during the interval, the difference between the 1915 distribution and 
the 1918 distribution in the United States, extreme as it is, cannot be said 
to be unreasonably or unbelievably great. Even the changes in the slope 
of the least-squares line are not phenomenal. Pareto's Prussian figures 
contain fluctuations in slope from — 1.60 to — 1.89 while the slope of the 
least-squares straight line fitted to his Basle data is only — 1.25. The 

' Between 12 and 13 times the average income (per capita) each year. 

2 Between 1,200 and 1,300 times the average income (per capita) each year. 



368 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

slopes of the least-squares straight lines fitted to the American data are 
—1.42 for 1915 and —1.69 for 1918. 

3. If the upper income ranges (or '' tails ") of income distributions 
were, when charted on a double log scale, closely similar in shape, 
would that fact justify the assumption that the lower income ranges 
were likewise closely similar? 

Before attempting to answer the above question, let us summarize the 
case we have just made against believing the "tails" significantly similar. 
We can then discuss how much importance such similarity would have 
did it exist. 

We have found upon examination that the approximation to straight 
lines of the tails of income distributions plotted on double log scales is 
specious; that the slopes of the fitted straight lines differ sufficiently to 
produce extreme variations in the relative number of income recipients 
in the upper as compared with the lower income ranges of the tails; 
that the upper and lower income ranges of the actual data for different 
times or places tell a similar story of extreme variation; and that the 
irregularities in shape of the tails of the actual data, entirely aside 
from any question of approximating or not approximating straight lines 
of constant slope, vary greatly from year to year and from country to 
country, ranging all the way from the irregularities of such distributions 
as the Oldenburg data, through the American data for 1914, 1915 and 1916 
to such an entirely different set of irregularities as those seen in the Amer- 
ican data for 1918^ 

At this stage of the discussion the reader may ask whether a general 
appearance of approximating straight lines on a double log scale, poor as the 
actual fit may be found to be under analysis, has not some meaning, some 
significance. The answer to this question must be that, if we were not deal- 
ing with a frequency distribution but with a correlation table showing a 
relationship between two variables, an approximation of the regression lines 
to linearity when charted on a double log scale might easily be the clue 
to a first approximation to a rational law; but that, on the other hand, ap- 
proximate linearity in the tail of a frequency distribution charted on a double 
log scale signifies relatively little because it is such a common charac- 
teristic of frequency distributions of many and varied types. 

The straight line on a double log scale or, in other words, the equation 
y = fea?"", when used to express a relationship between two variables, is, to 
quote a well-known text on engineering mathematics, "one of the most 
useful classes of curves in engineering." ^ In deciding what type of equa- 
tion to use in fitting curves by the method of least squares to data con- 

1 Compare Charts 28H, 28B, 28C, 28D and 28F. 

2 P. Steinmetz, Engineering Mathematics, p. 216. 



PARETO'S LAW 369 

cerning two variables the texts usually mention y = hx"^ as "a quite com- 
mon case." ^ A recent author writes, "simple curves which approximate 
a large number of empirical data are the parabolic and hyperbolic curves. 
The equation of such a curve is ?/ = ax^ [y = hx"^], parabolic for h positive 
and hyperbolic for h negative." - A widely used text on elementary 
mathematics speaks of the equation y= hx^ as one of ''the three funda- 
mental functions" in practical mathematics.^ The market for ''logarith- 
mic paper" shows what a large number of two-variable relationships may 
be approximated by this equation. Moreover this equation is often a 
close first approximation to a rational law. Witness "Boyle's Law." In- 
deed, sufficient use has not been made of this curve in economic discus- 
sions of two-variable problems. 

The primary reason why approximation to linearity on a double log 
scale has no such significance in the case of the tail of a frequency distribu- 
tion as it often has in the case of a two-variable problem is because of 
the very fact that we are considering the tail of the distribution, in other 
words, a mere fraction of the data. While frequency distributions which 
can be described throughout their length by a curve of the type y = bx"^ are 
extremely rare, a large percentage of all frequency distributions have tails 
approximating straight lines on a double log scale.^ It is astonishing how 
many homogeneous frequency distributions of all kinds may be described 
with a fair degree of adequacy by means of hyperbolas ^ fitted to the data 
on a double log scale. Along with this characteristic goes, of course, the 
possibility of fitting to the tails of such distributions straight lines approxi- 
mately parallel to the asymptotes of the fitted hyperbola. However we 
have by no means adequately described an hyperbola when we have 
stated the fact that one of its asymptotes is (of course) a straight line and 
that its slope is such and such. Had we even similar information con- 
cerning the other asymptote also, we should know little about the hyper- 
bola or the frequency distribution which it would describe on a double 
log scale. The hyperbola might coincide with its asymptotes and hence 
have an a7igle at the mode or it might have a very much rounded "top." 
Such a variation in the shape of the top of the hyperbola ^ would generally 
correspond to a very great variation in the scatter or "inequality" of the 
distribution as well as many other characteristics. 

1 D. P. Bartlett, Method of Least Squares, p. 33. 

2 J. Lipka, Graphical and Mechanical Computation, p. 128. 

3 C. S. Slichter, Elementary Mathematical Analysis, preface. 

« A very large percentage of the remainder have tails approximating straight lines on a 
natural x log y basis. 

6 N. B. Not a straight line on the double log scale, which is a so-called hyperbola on the 
natural scale, but a true conic section hyperbola on the double log scale. 

Charts 2SK and 28L (Earnings per Hour of 318,946 Male Employees in 1919) illustrate 
how excellent a fit may often be obtained by means of an hyperbola even though fitted only 
by selected points. A comparison of the least-squares parabola and the selected-points 
hyperbola on Chart 28K illustrates also the straight-tail effect. 

« Compare Karl Pearson's concept of "kurtosis.". 



370 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 




PARETO'S LAW 



371 



CHART 28 L 



■80,000 



■70,000 



-60,000 S 



50,000 S 



■40,000 s 



-20,000 




EARNINGS PER HOUR 

OF* 

318,946 MALE EMPLOYEES 

MOffTHLY l/fBO/? ffBV/£y\f, S£PT. /3/& 
<SCf]L^S- /TftTUfffil. 




EARNINGS PER HOUR IN DOLLARS 

30 .40 .50 .60 .70 .80 



Rough similarity in the tails of two distributions on a double log scale 
by no means proves even rough similarity in the remainder of the dis- 
tributions. Charts 28M, 28N, 280 and 28P illustrate both cumulatively 



100,000 



q 
lO.OOOJ 



- 1,000 o 




CHART 28 M 

CUMUIMIVE FREOUEnCYOISTRIBUTlOH 
RATES OF WAGES PER HOUR 

FOR 

72,291 MALE EMPLOYEES 

SLAUfiHTERinG & MEAT-PACKING INDUSTRY 
m THE U.S. m 1917. 

souea mmwof ifidoiismiiT/a.Mun/rtfst- 
iCA\ei Loganthmio, 



WAGES IN CENTS PER HOUR 

15 20 25 30 40 50 

' I \ 1 I ■ 



SO 



572 PERSONAL DISTRIBUTION OF INCOME IN U. S. 



-10,000 



1,000 s 

H 



100 w 

s 



CHART 28 N 



FREOUEMCY DISTRIBUTIOM 

RATES OF WAGES PER HOUR 

73,2ai MALE EMPLOYEES 

SlAUGHTERIMGaMEAT-PACKING II1DU5TRY 
IMTME 0.5. IM 1917. 

spi/fcs ei/s[/!ii of meofsTm^Tici. euuermzsi 
Scales Lqgarithmia 




WAGES IN CENTS PER HOUR 

15 20 25 30 40 50 

' ' ' ' ' 



and non-cumulatively on a double log scale two wages distributions whose 
extreme tails appear roughly to approximate straight lines of about equal 
slope. -^ Charts 28M and 28N are from data concerning wages per hour 
of 72,291 male employees in the slaughtering and meat-packing industry 
in 1917; " Charts 280 and 28P are from data concerning wages per hour 
of 180,096 male employees in 32 manufacturing industries in the United 
States in 1900.^ A mere glance at the two non-cumulative distributions 
will bring home the fact that while they show considerable similarity in 
the upper income range tails, they are quite dissimilar in the remainder 

1 The illustration shows only "rough similarity" in the extreme tails. However, there 
seems no good reason for believing that even great similarity in the tails proves similarity 
in the rest of the distribution. It certainly cannot do so in the case of essentially hetero- 
geneous distributions, such as income distributions. 

= Bureau of Labor Statistics, Bulletin No. 252. 

3 Twelfth Census of the United States (IPOO), Special Report on Employees and Wages, 
Davis R. Dewey. 



PARETO'S LAW 



373 



















CHART 280 


















CUMULATIVE FREttUENn DISTRIBUTION 








_^^_^^ 










RATES OF WAGES PER HOUR 
180.096 MALE EMPLOYEES 








-100,000 




^ 










3E MAMUFACTURIHG IMDUSTRICS 


















m THE U.S. IM 1900. 


















SOUK£ ■DC»tY-ie"C£n5U3 












X 


V 




5cales Loaanthmic 


-10,000 g 










\ 








o 
















-1,000 
















\^ 


-100 








WAGES 


IN 


CENTS 


PER HOUR 


^ 






10 


15 


20 


25 


30 


40 50 


60 70 60 90 100 



of the curves. Moreover, in spite of this similarity of tails, the slaughtering 
and meat-packing distribution has a coefficient of variation of 30.5 while 
the manufacturing distribution has a coefficient of 47.7. In other words, 
the relative scatter or "inequality of distribution" is more than one-and-a- 
half times as great in the manufacturing data as it is in the slaughtering 
and meat-packing data. Furthermore, no discussion and explanation of 
greater essential heterogeneity in the one distribution than in the other 
will offset the fact that the tails are similar but \hQ distributions are dif- 
ferent. There seems indeed to be almost no correlation between the slope 
of the upper-range tail and the degree of scatter in wages distributions. 
Some distributions showing extremely great scatter have very steep tails, 
some have not.^ The frequency curve for the distribution of income in 
Australia in 1915 is radically different from either the curve for the United 
States in 1910 constructed by Mr. W. I. King or the curve for the United 
States in 1918 constructed by the National Bureau of Economic Research. 



^ The tails of wage distributions have in general much greater slopes than those of the 
upper (i. e., income-tax) range f'l income distributions. This is an outstanding difference 
between the two distributions Pareto's conclusions with respect to the convex appearance 
of the curve for wages are consistent with curves showing number of dollars per income-tax 
interval traceable to wo:;es but not with actual wage distributions showing number of 
recipients per wage intel v'al. Distributions based upon income from effort and distributions 
based upon income from such sources (mostly profits and income from property) as yield the 
higher incomes seem to have tails the one as roughly straight as the other. Indeed many 
wage distributions have tails more closely approximating straight lines than do income-tax 
data. 



374 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 







CHART 28 P 






FREQUENCY DISTRIBUTION 

RATES OF WAGES PER HOUR 

180,096 MALE EMPLOYEES 






V 32 MAriOFACTURiriG IHDUSTRIES 
V p IN THE U. S. IN ISOO 

\<*> Scales Lo^arHhmic 










a\ 


-10,000 


^ 


v\^4 


-1,000 


— '^--/ 


^\5 


i 

-100 u 

< 
s 






\ 


01 

-10 1 

z 




^Ti 


I 




\ 




10 


WAGES IN CENTS PER HOUR 
15 20 25 30 40 50 60 70 ?0 90 100 



Yet all three curves have tails on a double log scale quite as similar as is 
common with income-tax returns.-^ 

From this discussion we may draw the corollary that it is futile to at- 
tempt to measure changes in the inequality of distribution of income 
throughout its range by any function of the mere tail of the income fre- 
quency distribution. It seems unnecessary therefore to discuss Pareto's 
suggestions on this subject. 

4. Is it probable that the distribution of income is similar enough 
from year to year in the same country to make the formulation 
of any useful general "law" possible? 

' As will be seen in Chapter 29, there seems reason for believing that the extreme difference 
between the distribution of incomes obtained by the Australian Census and the estimate 
made by the National Bureau of Economic Research is due largely to difference in definition 
of income and income recipient. However, this does not alter the fact that we have here 
again two distributions with tails as similar as is usual with income-tax distributions and 
lower ranges about as different as it is possible to imagine. 



PARETO'S LAW 375 

Before answering this question we must decide what we should mean 
by the word similar. If income distributions for two years in the same 
country were such that each distribution included the same individ- 
uals and each individual's income was twice as large in the second year 
as it had been in the first year, it would seem reasonable to speak of the 
distributions as strictly similar. If in a third year (because of a doubling 
of population due to some hypothetical immigration) the number of per- 
sons receiving each specified income size was exactly twice what it was 
in the second year, it would still seem reasonable to speak of the distribu- 
tions as strictly similar. Tested by any statistical criterion of dispersion 
which takes account of relative size (such as the coefficient of variation), 
the dispersion is precisely the same in each of the three years. Moreover 
the three distributions mentioned above ^ must necessarily have identically 
the same shape on a double log scale, and furthermore any two distribu- 
tions which have identically the same shape on a double log scale " must 
necessarily have the same relative dispersion as measured by such indices 
as the coefficient of variation, interquartile range divided by median, etc. 
Approximation to identity of shape on a double log scale seems then a 
useful concept of ''similarity." It is the concept implicit in Pareto's work.^ 

Now we have already found considerable evidence that income dis- 
tributions are not, to a significant degree, similar in shape on a double log 
scale. The income-tax tails of income distributions for different times and 
places neither approximate straight lines of constant slope nor approxi- 
mate one another; they are of distinctly different shapes. Moreover, such 
tails do not show in respect of their numbers of income recipients and 

1 Or, any distributions whose equations may be reduced to one another by substituting 
kiX for X and kiy for y. 

2 The curve may be thought of as consisting of two parts, which before reduction to log- 
arithms, would be (1) the positive income section and (2) the negative income section with 
positive signs. 

3 While approximate identity of shape on a natural scale, a natural x and log y scale, or 
any other similar criterion would constitute a "law," no such approximate identity of shape 
on such scales has yet been discovered and it seems difficult to advance any very cogent 
a priori reasons for expecting it. 

In this connection we must remember that had we the exact figures for the entire frequency 
curves of the distribution of income in the United States from year to year, if moreover we 
could imagine definitions of incom.e and income recipient which would be philosophically 
satisfactory and statistically usable — and if further we managed year by year to describe 
our data curves adequately by generalized mathematical frequency curves of more or less 
complicated variety we should not necessarily have arrived at any particularly valuable re- 
sults. Any series of data may be described to any specified degree of approximation by a 
power series of the type y = A -{- Bx -{- Cx'^ -\- Dx^ -{- . but such fit is purely em- 
pirical and absolutely meaningless except as an illustration of MacLaurin's theorem in the 
differential calculus. We might be able to describe each year's data rather well by one of 
Karl Pearson's generalized frequency curves, but if the essential characteristics of the curve — 
skewness, kurtosis, etc., changed radically from year to year, description of the data by such 
a curve might well give no clue whatever as to any "law." Not only might the years be dif- 
ferent but the fits might be empirical. Professor Edgeworth has well said that "a close fit 
of a curve to given statistics is not, per se and apart from a priori reasons, a proof that the 
curve in question is the form proper to the matter in hand. The curve may be adapted to the 
phenomena merely as the empirically justified system of cycles and epicycles to the planetary 
movements, not like the ellipse, in favor of which there is the Newtonian demonstration, as 
well as the Keplerian observations." Journal of the Royal Statistical Society, vol. 59, p. 533. 



376 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

total amounts of income any uniformity of relation to the total number 
of income recipients and total amount of income in the country, even 
after adjustments have been made for variations in population and average 
income.^ Considerations such as these, reenforce the conclusion which 
we arrived at from an examination of wage distributions, namely, that 
there is little necessary relation between the shape of the tail and the shape 
of the body of a frequency distribution, and have led us to suspect that, 
even if the tails of income distributions were practically identical in shape, 
it would be extremely dangerous to conclude therefore that the lower 
income ranges of the curves were in any way similar. 

A most important matter remains to be discussed. What right have 
we to assume that the heterogeneity necessarily inherent in all income 
distribution data is not such as inevitably to preclude not only uniformity 
of shape of the frequency curve from year to year and country to country 
but also the very possibility of rational mathematical description of any 
kind unless based upon parts rather than the whole? What evidence have 
we as to the extent and nature of heterogeneity in income distribution 
data? 

In the first place we must remember that lower range incomes are pre- 
dominantly from wages and salaries, while upper range incomes are pre- 
dominantly from rent, interest, dividends and profits.^ While 74.67 per 
cent of the total income reported in the United States in the $l,000-$2,000 
income interval in 1918 was traceable to wages and salaries, only 33.10 
per cent of the income in the $10,000-$20,000 interval was from those 
sources, and only 15.92 per cent of the income in the $100,000-$150,000 
interval and 3.27 per cent of the income in the over-$500,000 intervals. 
On the other hand, while only 1.93 per cent of the total income reported 
in the $l,000-$2,000 interval in 1918 was traceable to dividends, 23.73 
per cent was so traceable in the $10,000-$20,000 interval, 43.18 per cent 
in the $100,000-$150,000 interval, and 59.44 per cent in the over-$500,000 
intervals.^ The difference in constitution of the income at the upper and 

' Estimated per cent of total income received by highest 5% of income receivers in United 
States: 

1913 .33 

1914 32 

1915 32 

1916 34 

1917 29 

1918 26 

1919 24 

National Bureau of Economic Research, Income in the United States, vol. 1, p. 116. 
' Compare Professor A. L. Bowley's paper on "The British Super-Tax and the Distribution 
of Income," Quarterly Journal of Ecoyiomics, February, 1914. 
3 Statistics of Income 1918, pp. 10 and 44. 

While the reporting of dividends was almost certainly less complete in the lower than in 
the upper income classes, the difference could not be sufficient to invalidate the general con- 
clusion. Lower range incomes are predominantly wage and salary incomes; upper range in- 
comes are not. 



PARETO'S LAW 377 

lower ends of the distribution is sufficient to justify the statement that 
most of the individuals going to make up the lower income range of the 
frequency curve are wage earners, while the individuals going to make up 
the upper income range are capitalists and entrepreneurs.^ What do we 
know about the shapes of these component distributions? Is the funda- 
mental difference in their relative positions on the income scale their only 
dissimilarity? 

In any particular year the upper income tail of the frequency distribu- 
tion of income among capitalists and entrepreneurs seems not greatly dif- 
ferent from the extreme upper income tail of the frequency distribution 
of income among all classes. This is what we might expect. Not only is 
the percentage of the total income in the extreme upper income ranges 
reported as coming from wages and salaries small but much of this so- 
called wages and salaries income must be merely technical. For example, 
it is often highly "convenient" to pay "salary" rather than dividends. 
Furthermore, in so far as the tail of the curve of distribution of income 
among capitalists and entrepreneurs is not identical with the tail of the 
general curve, it will show a smaller rather than a larger slope, because the 
percentage of the number of persons in each income interval who are 
capitalists and entrepreneurs increases as we pass from lower to higher 
incomes.- Now the slopes of the straight lines fitted to the extreme tails 
of non-cumulative income distributions on a double log scale fluctuate 
within a range of about 2.4 to 3.0. 

The upper range tails of wages distributions tell an entirely different 
story. Aside from surface irregularities often quite evidently traceable to 
concentration on certain round numbers, the majority of wages distribu- 
tions have tails which, on a double log scale, are roughly linear.^ How- 
ever the slopes of straight lines fitted to these tails are much greater than 
the slopes of corresponding straight lines fitted to income distribution 
tails.^ While the slopes of income distribution tails range from about 2.4 

^ Many individuals in the middle income ranges must necessarily be difficult to classify. 
This does not mean that the concept of heterogeneity is inapplicable. There are countries 
in which the population is a mixture of Spanish, American Indian, and Negro blood. Now 
such a population must, for many statistical purposes, be considered extremely heterogeneous 
even though the percentage of the population which is of any pure blood be quite negligible. 

2 In 1917, the only year in which returns are classified according to "principal source of 
income" (wages and salaries, income from business, income from investment) the difference 
in slope, in the income range $100,000 to $2,000,000, between the distribution for all returns 
and the distribution for those returns which did not report wages and salaries as their prin- 
cipal source of income was less than .05. The slope in this range of the line fitted to all re- 
turns was about 2.64; the business and investment line was about 2.59 and the wages line 
about 3.21. In 1916, the only year in which returns are classified according to occupations, 
the distribution of income among capitalists shows a slope of only 2.08 while public service 
employees {civil) show a slope of 2.70 and skilled and unskilled laborers a slope of 2.74. 

3 Attention has already been drawn to the fact that this is a characteristic of many fre- 
quency distributions of various kinds. 

* A further difference between the upper range income distribution among capitalists and 
entrepreneurs and the upper range of the distribution among all persons seems to be, from 
the 1916 occupation distributions, that the distribution among all persons shows less of a roll, 
i. e., is straighter. 



378 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

to 3.0, the slopes of wages distributions tails commonly range between 
4.0 and 6.0. They seldom run below about 4.5; they sometimes run as 
high as 10.0 and 11.0. 

A distribution of wages per hour for 26,183 male employees in iron and 
steel mills in the United States in 1900 ^ shows a tail with a slope of about 
3.35. However, the total of which this is a part, the distribution of wages 
per hour among 180,096 male employees in 32 manufacturing industries 
in 1900, shows a tail-slope of about 4.8. The estimated distribution of 
weekly earnings of 5,470,321 wage earners in the United States in 1905 ^ 
shows a tail-slope of about 5.0. The distribution of earnings per hour 
among 318,946 male employees in 29 different industries in the United 
States in 1919 ^ shows a tail-slope of about 5.86. The distribution of 
wages per month among 1,939,399 railroad employees in the United States 
in 1917 ^ shows a tail-slope of about 6.25. The distribution of wages per 
hour among 43,343 male employees in the foundries and metal working 
industry of the United States in 1900 '" shows a tail-slope of about 7.8. 
The distribution of earnings in a week among 9,633 male employees in the 
woodworking industry — agricultural implements — in the United States in 
1900'' shows a tail-slope of over 11.0. At the other extreme was the case 
of the wages-per-hour distribution among 26,183 male employees in Amer- 
ican iron and steel mills in 1900 with a slope of 3.35. Both 11.0 and 3.35 
are exceptional, but the available data make it clear that wages distribu- 
tions of either earnings or rates have tail-slopes which are always much 
greater than the maximum tail-slope of income distributions. 

The illustrations in the preceding paragraph are illustrations of the tail- 
slopes of wages distributions among wage earners. However all the evi- 
dence points to frequency distributions of income among wage earners 
having tail-slopes only very slightly less steep than the tail-slopes of wages 
distributions. We have almost no usable data concerning the relation 
between individual wage distributions and income distributions for the 
same individuals, but we have a few samples showing the relation between 
family earnings distributions and family income distributions.'^ More- 
over, we can without great risk base certain extremely general conclusions 

'Twelfth Census of the United States (1900), Special Report on Employees aiid Wages, 
Davis R. Dewey. 

2 1905 Census of Manufacturers, Part IV, p. 647. 

3 Monthly Labor Review, Sept., 1919. 

* Report of the Railroad Wage Commission to the Director General of Railroads, 1919, p. 96. 

•i Twelfth Census of the United States (1900), Special Report on Employees and Wages, 
Davis R. Dewey. 

6 Twelfth Census of the United States (1900), Special Report on Employees and Wages, 
Davis R. Dewey. 

' The reader must not confuse the percentage of the income not derived from wages going 
to wage-earners in any particular income class with the percentage of the income not derived 
from wages going to all income recipients in any particular income class. Some of these last 
recipients are not wage earners at all, they receive no wages. Information concerning the 
second of these relations but not the first is given in the income tax reports. 



PARETO'S LAW 379 

concerning individual wage-earners' income distributions on these family- 
data. The upper tails of the family-wage distributions are the tails of the 
wage distributions for the individuals who are the heads of the families. 
This is apparent from an analysis of the samples. Now income from rents 
and investments belongs almost totally to heads of families. Such income 
is however so small in amount that it cannot alter appreciably the slope 
of the tail.^ While income from other sources than rents and investments 
(lodgers, garden and poultry, gifts and miscellaneous) may not be so con- 
fidently placed to the credit of the head of the family, this item changes 
its percentage relation to the total income so slowly as to be negligible in 
its effect upon the tail-slope of the distribution.- Notwithstanding the 
danger of reasoning too assuredly about individuals from these picked 
family distributions, we seem justified in believing that the tail-slopes of 
income distributions among individual wage earners are not very different 
from the tail-slopes of wage distributions among the same individuals.^ 
The upper tail-slopes of income distributions among typical wage earners 

^ For example, in the report on the incomes of 12,096 white families published in the Monthly 
Labor Review for December, 1919, we find the income from rents and investments less than 
one per cent of the total family income for each of the income intervals. 

Percentage income from 

Income group rents and investments 

is of total income 

Under $900 .079 

$ 900-$l,200 .176 

1,200- 1,500 .410 

1,500- 1,800 .551 

1,800- 2,100 .606 

2,100- 2,500 .998 

2,500 and over . 778 _ 

- As a somewhat extreme example, the Bureau of Labor investigation mentioned in the 
preceding note shows the following relations between total family earnings and total family 
income (including income from rents and investments, lodgers, garden and poultry, gifts and 
miscellaneous) . 

Income group Percentage that total 

^ ^ earnings are of total income 

Under $900 96.2 

$ 900-$ 1,200 96.5 

1,200- 1,500 96.3 

1,500- 1,800 96.0 

1,800- 2,100 96.3 

2,100- 2,500 95.1 

2,500 and over 96.2 

3 Further corroboratory evidence, of some slight importance, that the tail-slopes of wage 
distributions among wage earners are not very different from the tail-slopes of income dis- 
tributions among wage earners is yielded by the fact that the tail-slopes of income distribu- 
tions among families (which are virtually identical with the tail-slopes of both income and 
wage distributions among the heads of these families) have roughly the same range as the 
tail-slopes of wage distributions among individuals. The British investigation into the in- 
comes of 7,616 workingmen's families in the United States in 1909 shows a tail-slope of about 
3.5. (Report of the British Board of Trade on Cost of Living in American Towns, 1911. [Cd. 
5609], p. XLIV.) The Bureau of Labor's investigation into the income of 12,096 white fam- 
ilies in 1919 shows a tail-slope of about 4.0. Mr. Arthur T. Emery's extremely careful in- 
vestigation into the incomes of 2,000 Chicago households in 1918 shows a tail-slope of 
about 4.4. At the other extreme we find that the Bureau of Labor's investigation into the 
income of 11,156 families in 1903 {Eighteenth Annual Report of the Commissioner of Labor, 
1903, p. 558) shows a tail-slope of about 10.0, and that Mr. R. C. Chapin's investigation into 
the income of 391 workingmen's families in New York City {Standard of Living A?nong Work- 
ingmen's Families in New York City, p. 44) also shows a slope of about 10.0. The tails of 
these last two cases are very irregular so that the slope itself is not determinable with much 
precision. 



380 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

may then be assumed to have much greater slopes than the upper tail- 
slopes of mcome distributions among capitalists and entrepreneurs. It 
does not seem possible to make any very definite statement concerning 
the body and lower tail of the capitalist and entrepreneurial distribution — 
even in so far as that term is a significant one.^ All the evidence suggests 
that the mode of what we have termed the capitalist-entrepreneurial dis- 
tribution is consistently higher than the wage-earners' mode.^ Its lower 
income tail undoubtedly reaches out into the negative income range, which 
the tail of the wage-earners' distribution may, both a -priori and from evi- 
dence, be assumed not to do. It seems a not irrational conclusion then to 
speak of the capitalist-entrepreneurial distribution as having a lesser tail- 
slope than the wage- earners' distribution on the lower income side as well 
as on the upper income side,^ and as a corollary almost certainly a much 
greater dispersion both actual and relative than the wage-earners' dis- 
tribution. 

Though the above generalizations concerning differences between the 
wage-earners' income distribution and the capitalist-entrepreneurial in- 
come distribution seem sound, they tell but a fraction of the story. Aside 
from the difficulty of classifying all income recipients in one or the other 
of these two classes, we are faced with the further fact that investigation 
suggests that our two component distributions are themselves exceedingly 
heterogeneous.'* We have already noted that wage distributions for dif- 
ferent occupations and times are extremely dissimilar in shape and we 
suspect that the same applies to capitalist-entrepreneurial distributions. 
For example, what little data we possess suggest that the distribution of 
income among farmers has little in common with other entrepreneurial 
distributions. 

Moreover, the component distributions, into which it would seem nec- 
essary to break up the complete income distribution before any rational 
description would be possible, not only have different shapes and different 
positions on the income scale (i. e., different modes, arithmetic averages, 
etc.), but the relative 'position with respect to one another on the income scale 
of these different component distributions changes from year to year.^ 

^ In the total income curve there is a broad twilight zone where individuals are often both 
wage or salary earners and capitalists or even entrepreneurs. 

2 In the 1916 occupation distributions the only occupations showing more returns for the 
$4,000-$5,000 interval than the .1i3,000-$4,000 (that is the only occupations showing any 
suggestion of a mode) are of a capitalistic or entrepreneurial description — bankers; stock- 
brokers; insurance brokers; other brokers; hotel proprietors and restaurateurs; manufacturers; 
merchants; storekeepers; jobbers; commission merchants, etc.; mine owners and mine op- 
erators; saloon keepers; sportsmen and turfmen. 

3 Of course the very word slope is an ambiguous term to use concerning the tail of a curve 
which enters the second quadrant. 

^ Evidence suggesting definite heterogeneity in the "wage and salary" figures of the income- 
tax returns is presented in Chapter 30. 

6 This fact is one of the simpler pieces of evidence against the existence of a "law." Of 
course, even though the income distribution were made up of heterogeneous material, if the 



PARETO'S LAW 



381 



Table 28Q ^ is interesting as showing the changes in the relative positions 
of the arithmetic averages of different wage distributions in 1909, 1913 
and 1918. 

TABLE 280 

CHANGES IN THE RELATIVE POSITIONS OF THE AVERAGE ANNUAL 
EARNINGS OF EMPLOYEES ENGAGED IN VARIOUS INDUSTRIES 



Industry 


1909 


1913 


1918 


All Industries 


100.0 

48.2 
95.7 

91.2 
111.7 
104.9 

104.0 

99.5 
123.5 
123.0 
118.1 
114.4 


100.0 

45.4 

104.4 

97.5 
103.5 
105.4 

108.2 

93.8 
114.1 
128.6 
113.8 
107.7 


100 


Agriculture 


54 7 


Production of Minerals 

Manufacturing: 

Factories 


119.0 
103 5 


Hand Trades 


110 8 


AU Transportation 

Railway, Express, Pullman, 

Terminal Cos 

Street Railway, Electric Li 

Telegraph and Telephone 
Transportation by Water. . 


Switching and 

^ht and Power, 
Cos 


119.3 

129.3 

81.4 
147 5 


Banking 

Government 

Unclassified Industries 


135.5 

83.0 

97.8 



The data are so inadequate that the construction of a similar table for 
capitalist-entrepreneurial distributions is not feasible. However, there are 
comparatively^ good figures for total income of farmers and total number 
of farmers year by year.- The average incomes of farmers, year by year, 
were the following percentages of the estimated average incomes of all 
persons gainfully employed in the country. 

Percentages 

1910 75.19 

1911 69.13 

1912 72.41 

1913 74.88 

1914 76.33 

1915 80.45 

1916 82.85 

1917 104.51 

1918 109.68 

1919 103.95 

1920 63.88 

This is a wide range. 

Exactly what effects have such internal movements of the component 
distributions upon the total income frequency distribution curve? This 
is a difficult question to answer as we have not sufficient data to break 

component parts remained constant in shape and in their relative positions with respect to one 
another on the income scale, these relations would of themselves constitute a "law" 

^ Based upon Income in the United States, Vol. I, pp. 102 and 103. 

2 See Income in the United States, Vol. I, p. 112. 



382 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



down the total, composite, curve into its component parts with any de- 
gree of confidence.^ However, the movements of wages in recent years 
would appear to give us a clue to the sort of phenomena we might expect 
to find if we had complete and adequate data. 

The slopes of the upper income tails of wages distributions are great, 
4 to 5 or more." Now the wage curve moved up strongly from 1917 to 
1918 if we may judge by averages. The average wage of all wage earners 
in the United States ^ increased 15.6 per cent "* from 1917 to 1918. During 
the same period the average income of farmers increased 19.1 per cent *" 
and the average income of persons other than wage earners and farmers 
remained nearly constant. Total amounts of income by sources in millions 
of dollars were: 





1917 


1918 


Percentage 1918 
was of 1917 


Total Wages « 


$27,795 

8,800 

17,265 


$32,575 
10,500 
17,291 


117.20 


Total Farmers' Income 

All other Income 


119.32 
100.15 






Total Income 


$53,860 


$60,366 


112.08 







o Includes pensions, etc., and includes soldiers, sailors, and marines. 

Stockholders in corporations saw income from that source actually decline 
from 1917 to 1918.^ What happened to American income-tax returns 
during this time? 

' The processes by which the income distribution curve published in Income in the United 
States, Vol. I, pp. 132-135 was arrived at were such that to use that material here would 
practically amount to circular reasoning. The conclusions arrived at here were used in build- 
ing up that curve. 

~ The slope of the tail of the wage and salary curve in the 1917 income tax returns is only 
about 3.21 (compare, note 2, p. 377). However we must remember that the individuals there 
classified are largely of an entirely different type of "wage-earner" from those in the lower 
groups. In this upper group occur the salaried entrepreneurs, professional men, etc., and 
those whose "salaries" are really profits or dividends. The evidence points to a rather dis- 
tinct and significant heterogeneity along this division in the wage and salary distribution. 
See Chapter 30. 

3 Excluding soldiers, sailors, and marines, and professional classes but including officials 
and "salaried entrepreneurs." 

< From $945 per annum in 1917 to $1,092 per annum in 1918. 

5 From $1,370 per annum in 1917 to $1,632 per annum in 1918. 



6 CORPORATION DIVIDENDS, SURPLUS AND EARNINGS 

(In millions of dollars) 




Dividends 


Surplus 


Net earnings 


1917 


3,995 
2,568 


3,963 
1,945 


7,958 


1918 


4,513 



See page 324. 



PARETO'S LAW 



383 



TOTAL AMOUNT OF NET INCOME RETURNED BY SOURCES (RETURNS 
REPORTING OVER $2,000 PER ANNUM NET INCOME) a 



(Millions of dollars) 



Income class 



Wages and salaries 



All other sources & 



1917 


1918 


1917 


1918 


$3,648 


$6,493 


$7,543 


$7,198 


1,553 


3,687 


1,799 


2,036 


301 


703 


528 


736 


661 


849 


1,167 


1,296 


1,133 


1,254 


4,049 


3,130 



Over $2,000. . 

2,000- 4,000. 

4,000- 5,000. 

5,000-10,000. 
Over 10,000. 



« Wages income from returns reporting between $1,000 and $2,000 per annum is not avail- 
able for 1917. 

& "Other sources" are total net income minus wages and salaries, i. e., total general deduc- 
tions have been assumed as deductible from other sources (gross). All things considered, 
this seems proper here though it may easily be criticised. In connection with changes in the 
relation between net and gross income from 1917 to 1918 see Chapter 30, pp. 401 and 402. 

While reported income from all other sources than wages and salaries 
declined 4.6 per cent,^ reported income from wages and salaries increased 
78.0 per cent.^ Moreover, the great increases in wages and salaries were 
in the lowest intervals. The wage curve with its steep tail-slope was 
moving over into the income tax ranges.^ The effect upon the total curve 
is very pronounced, as may be seen from Table 28R. 

TABLE 28R 



AMERICAN INCOME TAX RETURNS IN 1917 AND 1918 



Total Number of Returns 
(In thousands) 



1917 



1918 



Percentage 1918 
was of 1917 



$2,000-$4,000 
4,000- 5,000 
5,000-10,000 

Over 10,000. 



1,214 
186 
271 
162 



2,107 
322 
319 
160 



173.56 
173.12 
117.71 

98.77 



On a double log scale we see the curve changing its shape radically. While 
the 1917 curve is comparatively smooth and regular, the 1918 curve 
develops a distinct "bulge" in the lower ranges.^ 

The preceding discussion has been concerned with equal dollar-income 

^ Had "other sources" been taken gross instead of net, that item would have shown an 
increase of 5.3 per cent instead of a decrease of 4.6 per cent. 

2 The actual spread is still greater than the figures show. Income from professions, which 
in 1917 was classed under wages, in 1918 and 1919 was classed under business. 

3 This seems to be a fact though it is not the whole story. The "intensive drive" of 1919 
may easily account for some of the increase. See Chapter 30 for a discussion of the probable 
extent of this influence. 

* See Income in the United States, Vol. I, Charts 28 and 30. 



384 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



intervals. However, $2,000 income in 1918 was relatively less than $2,000 
income in 1917. The average (per capita) income of the comitry was 
$523 in 1917 and $586 in 1918.^ The adjustment is theoretically crude, 
but $2,241 ^ in 1918 might be considered as in one sense equivalent to 
$2,000 in 1917. The results of comparisons of the two years upon this 
basis are given in Table 28S.^ 

TABLE 28S 



INCOME RETURNED— BY SOURCES 

(Millions of dollars) 
1917 



Income class 


Wages and 
salaries 


Total net 
income 


Total net 

income 

minus 

wages and 

salaries 


Total grross 
income 


Total ijross 

income 

minus 

wages and 

salaries 


$2,000-$4,000. . .. 

4,000- 5,000. ... 

5,000-10,000. . .. 
Over 10,000.... 


$1,553 

301 

681 

1,133 


$3,352 

829 

1,828 

5,182 


$1,799 

528 

1,167 

4,049 


$3,713 

895 

1,951 

5,518 


$2,161 

594 

1,290 

4,384 



1918 



$2,241-$4,482. . .. 


$3,236 


$5,359 


$2,123 


$5,766 


$2,530 


4,482- 5,602. . .. 


498 


1,111 


613 


1,247 


749 


5,602-11,205. .. . 


773 


1,960 


1,187 


2,315 


1,542 


Over 11,205 


1,153 


4,129 


2,976 


4,842 


3,689 



(Multiplied by — , that is reduced to " 1917 dollars") 



$2,241-$4,482. ... 


$2,888 


$4,783 


$1,895 


$5,146 


$2,258 


4,482- 5,602. ... 


445 


992 


547 


1,113 


668 


5,602-11,205. . . 


690 


1,749 


1,059 


2,066 


1,376 


Over 11,205.... 


1,029 


3,685 


2,656 


4,321 


3,292 



(Percentages of Total Income of Country) 
1917 



$2,000-$4,000 . . 

4,000- 5,000. . 

5,000-10,000. . 
Over 10,00. . . 



2.88 

.56 

1.23 

2.10 



6.22 
1.54 
3.39 
9.61 



3.34 

.98 
2.16 
7.51 



6.89 

1.66 

3.62 

10.24 



4.01 
1.10 
2.39 
8.14 



1918 



$2,241-$4,482 .... 


5.30 


8.78 


3.48 


9.45 


4.15 


4,482- 5,602. . .. 


.82 


1.82 


1.00 


2.05 


1.23 


5,602-11,205. . .. 


1.27 


3.21 


1.94 


3.80 


2.53 


Over 11,205.... 


1.89 


6.77 


4.88 


7.94 


6.05 



1 Income in the United States, Vol. I, p. 76. 

2 $2,000 X — • 

' The figures for the amounts of income in the irregular 1918 income intervals of that table 
($2,241-$4,482, etc.) were calculated by straight line interpolation on a double log scale ap- 
plied to the even thousand dollar intervals of the income-tax returns. Though the total 
income curve docs not approximate linearity it may be assumed linear within the small 
range of one income tax interval without serious error. 



PARETO'S LAW 



385 



(Table 28S concluded.) 



NUMBER OF RETURNS 

(Thousands) 



Income class 


1917 


Income class 


1918 


Percentage 1918 
was of 1917 


$2,000-14,000 

4,000- 5,000 

5,000-10,000 

Over 10,000 


1,214 
186 
271 
162 


$2,241-$4,482 

4,482- 5,602 

5,602-11,205 

Over 11,205 


1,758 
220 
260 
136 


144.81 

118.28 

95.94 

83.95 



It is from this table once again apparent that the wage distribution moved 
independently up on the income scale and that the effect of this movement 
was confined to the lowest income intervals. Charts 28T, 28U, 28V, 28W, 
28X, 28Y, 28Z, and 28AA which show the number of dollars income per 
dollar-income interval, by sources, are enlightening as illustrating in still 



CHART 28 T 



U.S. INCOME TAX RETURNS 
I9IS 
NUMBER Of DOLLARS !M EACH INCOME 
INTERVAL BY SOURCES 
Scales Logarithmic. 
1. TOTAL INCOME. 
Z WAGES. 

3. BUSINESS 

4. OTHER INCOME 



10,000 



1,000 u 




INCOME IN THOUSANDS OF DOLLARS 
20 30 40 50 lOO 200 



300 400 500 



1,000 



2,000 



386 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



CHART 28d 



U S. IHCOME TAX RETURNS 

1916 

NUMBER OF DOLLARS m EACH 

IMCOME IMTLRVAL BY SOURCES. 

Scales Logarithmic. 

A INCOME OTHER THAH WAGES 

OR BUSIME5S. 
5 REHT5 
G. INTEREST 
7 DIVIDEMD3. 




INCOME IN THOUSANDS OF DOLLARS 
20 30 40 50 100 200 



300 400 500 



1.000 



2.000 



PARETO'S LAW 



387 



CHART 28V 



-100.000 



-10,000 



- 1,000 ^ 



U.S. INCOME TAy RETURNS. 
1917 

NUMBER OF DOLLARS IM EACH 
INCOME INTERVAL BY SOURCES. 

Scales Logarithmic. 

1. TOTAL INCOME. 
8 WAGES. 

3. BUSINESS. 

4. OTHER INCOME. 




V-^3 



INCOME IN THOUSANDS OFDOLLARS 

20 30 40 50 100 200 



300 400 500 



1,000 



388 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 




PARETO'S LAW 



389 



f"'' ^N 


S.^^^ 




CHART 28 X 


1,000,000 

,» 


^•n\ 




U S. INCOME TAX RETURNS 




^^ 




1918 


ji^^''^'^'^"^ 


V 


NUMBER Of DOLLARS IN EACH 


r 


^Sv^ 


mCOME INTERVAL BY SOURCES 




^■^^ 


^Ni^,^^ 


Scales Logarithmic 


100,000 




N^\,,^^^ 


1 TOTAL INCOME 

2 WAGES 

3 BUSINESS 






*r;v^>i„^^ ^\^ 


4 OTHER INCOME. 


10.000 

!,oaog 








H 
Z 






Vxx-x 









\ "^ -. ^x^x 


z 






""^-v ""^N. ^v\ 


130 as 






*\ \ ^^Ni^ 


3 

-J 

§ 






\ \ ^^ 


i 






\. " "^ \ ^i 


10 S 

<: 






-X \ 


^ 
u 






> \ 











a 
1 




• 








INCOME IN THOUSANDS OF 
10 20 30 40 50 


DOLLARS \ 


2 3 


4 5 


100 200 300 400 500 1,000 2,000 3,000*2 



390 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



U. 5. IMCOME TAX RETURN5 
I9IS 
NUMBER OF DOLLARS IM EACH 
inCOME INTERVAL BY 50URCE5 

.Scales Logarithmic. 
4. INCOME OTHER THAN WAGES 

OR BU51NE55. 
5 RENTS 

6, INTEREST. 

7. DIVIDENDS 




INCOME IN THOUSANDS OF DOLURS 
20 30 40 50 100 



200 300 400 500 



PARETO'S LAW 



391 



-1,000 g 





CHART 28 Z 




US INCOME TAX RETURNS 


' " — <\ ^s. 


1919 


-^ 


NUMBER OF DOLLARS IN EACH 


INCOME INTERVAL BY SOURCES 


,,^j^^ Scales Logarrthm'ic. 


^Sv^ 1. TOTAL INCOME. 




^>v 2 WAGES 




». >v 3. BUSINESS 




^^^^„,^ >^^^ A. OTHER INCOME. 










X \N. 




\ x^\». 




^^ \^\\ 




'\ ^^^!vv 




\ '>\ ^v 




\^ \0v— ' 




> ^r- — "^ 




\ ^ -' 




\ / \ 




\ 
\ 

\ 
\ 

\ 



\ .2 



INCOME IN THOUSANDS OF DOLLARS 



200 300 400 500 



392 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



-10,000 



-10 S 



U S INCOME TAX RETUPM5 

1319 

NUMBER OF DOLLARS IN EACH 

INCOME IMTERVAL 5Y SOURCES 

Scales Lqgari'thmic 

INCOME OTHER THAN VYASES 

OR BUSINESS 
RENTS. 
INTEREST. 
DIVIDENDS. 




INCOME IN TH0US.4NDS OF D0LL.4RS 



20« .300 400 .WO 



PARETO'S LAW 393 

greater detail the changes in the constitution of the returns from year to 
year. 

Such material and the appearance of the "bulge" on the income-tax 
curve in the lowest income ranges ^ in the years 1918 and 1919 when wages 
and salaries were high and average (per capita) incomes also high ^ strongly 
suggest that the income curve, in so far as it shows any similarity from 
year to year, changes its general appearance and turns up (on a double 
log scale) as it approaches those ranges where wages and salaries are of 
predominant influence.^ The great slopes of wage distributions are on 
this hypothesis not inconsistent with the smaller slope of the general 
income curve in its higher (income-tax) ranges.'* 

Conclusions : 

(1) Pareto's Law is quite inadequate as a mathematical generahzation, 
for the following reasons: 

(a) The tails of the distributions on a double log scale are not, 
in a significant degree, linear; 

(b) They could be much more nearly linear than they are without 
that condition being especially significant, as so many dis- 
tributions of various kinds have tails roughly approaching 
linearity; 

(c) The straight lines fitted to the tails do not show even approxi- 
mately constant slopes from year to year or between coun try 
and country; 

(d) The tails are not only not straight lines of constant slope but 
are not of the same shape from year to year or between 
country and country. 

(2) It seems unlikely that any useful mathematical law describing the 
entire distribution can ever be formulated, because: 

(a) Changes in the shape of the income curve from year to year 
seem traceable in considerable measure to the evident hetero- 
geneity of the data; 

(b) Because of such heterogeneity it seems useless to attempt to 

1 See Chapter 30 for further discussion of this "bulge" in connection with an examination 
of how far it may be the result of irregularity in reporting. 

2 Average (per capita) incomes being high means that a definite money income (such as 
$2,000) takes us relatively further down the income curve than if average incomes were low. 

3 It is difficult to say just where the "bulge" might have appeared in the 1917 distribution 
if as great efforts had been made to obtain correct returns in that year as were made under 
the "intensive drive" for 1918 returns. The wages line on the 1917 number of dollars income 
per dollar-income interval chart (Chart 28V) shows signs of turning up somewhere between 
$4,000 and $5,000 and the business line somewhere in the $5,000-$10,000 interval. However 
neither movement is large nor can their positions be accurately determined on account of the 
size of the reporting intervals. See also Chapter 30, p. 412. _ 

*The " bulge" on the income from wages and salaries curve itself, as seen in the income- 
tax returns for 1918 and 1919 (see Charts 28X and28Z), seems the result of heterogeneity in 
these wage aud salary data themselves. This hypothesis is considered in Chapter 30. 



394 PERSONAL DISTRIBUTION OF INCOME IN TJ. S. 

describe the whole distribution by any mathematical curve 
designed to describe homogeneous distributions (as any simple 
mathematical expression must almost necessarily be designed 
to do) ; 

(c) Furthermore, the existing data are not adequate to break up 
the income curve into its constituent elements; 

(d) If the data were complete and adequate we might still remain 
in our present position of knowing next to nothing of the 
nature of any "laws" describing the elements.^ 

(3) Pareto's conclusion that economic welfare can be increased only 
through increased production is based upon erroneous premises. 
The income curve is not constant in shape. The internal movements 
of its elements strongly suggest the possibility of important changes 
in distribution. The radically different mortality curves for Roman 
Egypt and modem England,^ and the decrease in infant mortality 
in the last fifty years illustrate well what may happen to heteroge- 
neous distributions. 

The next four chapters review the data from which any income frequency 
distribution for the United States must be constructed. 

^ Though all the evidence points to hope of further progress lying in the analysis of the 
parts rather than in any direct attack upon the unbroken heterogeneous whole. 
2 See Biometrika, Vol. I, pp. 261-264. 



CHAPTER 29 
OFFICIAL INCOME CENSUSES 

There has never been a complete income census of the American people. 
The Federal income-tax data cannot take the place of such a census. Re- 
specting the distribution of income among persons having incomes of less 
than $1,000 Federal income-tax data give us no information whatsoever. 
Furthermore, on account of the exemption of married persons, compara- 
tively little use can be made of the $1,000 to $2,000 interval. The number 
of persons reporting incomes over $2,000 in our best year, 1918, was only 
7.3 per cent of the estimated total number of income-recipients in the 
country. Moreover, not only because of direct evasion and illegal non- 
reporting, but also because of "legal evasion" and the large amount of 
tax-exempt income which need not be reported at all, these income-tax 
data cannot give an approximately correct picture of even that part of 
the frequency curve which lies above $2,000. The adjustments of the 
income-tax data necessary to obtain such a picture are extremely large, 
as we shall presently see. 

Only one country in the world has ever taken an official income census 
which made any pretense of completeness. Under the War Census Act 
the Commonwealth of Australia took an official income census of incomes 
received during the year ended June 30, 1915, by everyone, man, woman, 
or child, who was ''possessed of property, or in receipt of income." ^ The 
results of that census are summarized by G. H. Knibbs, the Commonwealth 
Statistician, in The Private Wealth of Australia and its Growth. A Re- 
port of the War Census of 1915. (See Table 29A and Charts 29A, 29B 
and 29C.) 

Now while it would naturally be impossible to construct a complete 
frequency distribution for American incomes from AustraUan data,^ we 
might perhaps hope to discover some characteristics of income-distribution 

' While the first clause of the Australian "Wealth and Income Card" stated merely that 
it was "to be filled in by all persons aged 18 or upwards possessed of property, or holding 
property on trust, or in receipt of income," etc. (p. 9), "a special instruction was issued that 
in the case of all persons under the age of 18, possessed of property, or in receipt of income, 
a return must be furnished by the parent or guardian in respect of such property or income." 
(p. 10.) The income from such trust funds was not all, but only "in the main," allocated to 
individual beneficiaries, (p. 22.) 

G. H. Knibbs, The Private Wealth of Australia and its Growth. A Report of the War Census 
of 1915. 

2 Aside from the questionableness of such a procedure, the large size of the low income 
intervals in the Australian distribution and the lack of information concerning the amount 
of negative income make that distribution a difficult one to work with. A classification by 
such large intervals tells very little. 

395 



^^ 

So 

OS 

^> 
^^ 

O 
O 

l-H 



O 


Average 
income « 
(Pounds) 




© 

T-H 

T-H 


Amount 
of income 
(Nearest 
thousand 
pounds) 


OOO'-i'-H'Ml^OfOfOO'-tO^O 
OOfM-^i-OCOCOCOCOOOOOfMT-HTHiO 
00 I> CO O ^ OO^OO {M_^CO^O__!rO_0_c£5^01^C2_ 

o"'-o"^'~^>^c^^-'~o"'-^co"t-.'"^'~Tt^~(^^'-^~^>^ 

7-H CO O (N (M C<) 1-H 


CO 

CO 

T-H 

o" 

(M 




OLQr-('*--<t^tOiOCiGOOOCDTH'-ICO(M 
COO^tMCO-^iMCvJrHLOCOOiMOCOCO 

C2 T-< o !-< (M o^cc^r-_^«3_-^-'^G0_'*_o^i-^ ■* GO 

lO"^-"lO"I-^"o''o'"l>■"LO"oO~I>■'"lO~(^^C<^ 

<-H -+I cr. o lo t^ l-H lo 1-1 

CO Tfl -^ LO 1— 1 T-H 


T-H 


S 


1 

Average 
income « 
(Pounds) 


(MOOOOCO'^OiVDr^t>>'+'Oi-^t:^iCt^ 

C^0i-I0t^^t-0^0(MCOC01LO<M 

T-H tH T-H C-1 CO O 00 C^T,t^'* '^''^'^.'^^ 

t-rT-H'~c<rco'"'*"'i>" 




Amount 
of income 

(Nearest 
thousand 

pounds) 


t^OOOTT-HT-HOOCOOCTsOtMT-HOCCiD 
T-H,-HL0L0^-*CiC0l:^00iMl>-Oi0i-0 
1> Tt< (M LO CI O >* CO C5 0_CO I> CO tM O 

co"T-rcr c^fc^-c^fT-T T-p 

I-H 


T-H 

CO 
CO_^ 

oo" 

CO 


3 

S 


cOtMcOOi'-Hlr^T-Hl^.T-HLOio-^IXMGOO 

t^OOC^ILOOOTHOT^OOr-HOlOOO 
^ LO rH O CO CD O^CO_^CO__I-H_^a3 CO CO t-H 

orT-H''co'~(M'"co"cKrT^"co'"c<rr-r 

^ O CO lO T— I r-H 

C^ CO T-H 


CO 

00 


CO 


Average 
income " 
(Pounds) 


05Tt<C'0(Mcoi>-Tt<eo-^io-*T-HC<iTtico 

Cqt>(NiOt--COI>-0>OT-HCqcOT-H02GO 

tHt-h,-h(mcocooo |^l,l> ^^"*,'*^i>, 

T-TT-H^cfco^-^'or 


CO 
l-H 


Amount 
of income 
(Nearest 
thousand 
pounds) 


COOOOCOCiT-IQOCOCO-^cOOOiCO 
COOClCiT-HClOOOClOt^-^^OOO 
T-H CO O O^C^^T-H^^CO^O^CO^^O^CD^T-t^^Cvl^CO^^CO^^ 
■i:iH"Tt^~io'~l>.'~t^">--0"c/r05"L0'i0~C0'~Ttrc<l T-H !>. 
(MO (M (M T-H 


CO 
o 

o 

(M 


S 


OCOiQLOOO-^OOOOCOCOiMt^OiiOcO 
cDt-hc0Oc0»0(MO!Mt-(C0C0O'0I:^'*' 
■r}< lO 00 >-< CO CO CO T-H^C1^CO_0_T-H^1>-_^CO CO 1> 

co'~>o"t^'"oo''co'')f-ro'~ai'io"cir-^"(N T-H 

CO'*'(N^^lOO'*rH 
I-H CO ■* T-H T-H 


oo 

o 
oo 

CO__ 


CD 

i 

o 

a 
1— t 


OOcOOOOOOOOOOO 
OiOOOOOiOOOOOOO 
^T-H,-H(Mc0»0l>O»Ci 0^0_0_0_ 

■ ^ T-rT-rc^fco"-* LQ 




-do ^ o 


"o OoOOcOOOOOOOOOOO 
yH'OioOiOiOOOOiOOOOOOO 

oj c T-H r-i I-H (M CO ":n> o Lo o o o o 

Q P ^ i-h"t-h~(m'~CO'"'*~>0~ 



ti 

^ o 

"S Pi 
o g 



is 

a; bc 

S JH 



cC 



AUSTRALIAN CENSUS OF INCOMES - 1915. 




398 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

curves in general from this, the only actual census ever taken. A knowl- 
edge of such general characteristics might then, quite imaginably, be a 
little helpful in the problem of describing the American or any other 
income distribution. 

However, when we come to examine the Australian figures, we find that 
they have certain pronounced peculiarities which would be extremely diffi- 
cult to read into the American material. For example, the Australian dis- 
tribution shows a flatness and lack of pronounced mode totally unlike the 
results we have built up from an analysis of American data. In the Aus- 
tralian distribution there are nearly the same number of persons having 
incomes between and £50, £50 and £100, and £100 and £150.^ 

What are the causes of this rather startling peculiarity of the Australian 
frequency curve? ^ In the first place let us suggest a possibly minor but 
by no means necessarily negligible factor. We know little about the good- 
ness of the Australian reporting in this census. Income is, from its nature, 
a difficult subject to investigate. When the material is collected by means 
of schedules to be filled in by the informants, as was the case in the Aus- 
tralian census, the returns may easily be full of errors. The average in- 
dividual is surprisingly ignorant concerning the amount of his total income. 
The further fact that the census was taken in order to estimate possi- 
bilities of future taxation may well have been a powerful incentive towards 
great irregularities all along the line, but especially in the lower income 
groups. Persons whose income brought them distinctly into the upper 
groups (over £156) were, at the time of the income census, about to make 
returns under oath for income-tax purposes and would hardly care to 
show a radical discrepancy between the two returns. On the other hand, 
many persons, whose true incomes were around £156 and the modal income, 
might easily have ''underestimated" with the idea of evading if possible 
future taxation based upon a lowering of the exemption limit. The result 
of such practices would tend to show up graphically in a flattening of the 
curve in the vicinity of the mode of the distribution and a raising of the 
numbers in the lowest groups. ^ 

However, poor reporting is probably only a secondary element ac- 
counting for the peculiarities of the AustraUan curve. It is most of all the 

1 See Table 29A and Chart 29A. 

2 Notwithstanding the fact that distributions for different times and for different countries 
probably vary greatly (see Chapter 28), the difference between the Australian curve and 
the Bureau's American estimate seems too radical to explain upon this basis. 

2 It is difficult to determine the extent of actual non-reporting. The number of males 
filling out income cards was 2,527,831. All males "possessed of property, or in receipt of 
income" are supposed to be included in this number. It amounted, however, to only 54.60 
per cent of the total male population. Males "possessed of property, or in receipt of income" 
necessarily constitute a larger percentage of the total male population than do male "bread- 
winners," yet in the Australian census of 1911 male breadwinners constituted 69.4 per cent 
of the total male population, and male breadwinners 20 years of age or older 58.9 per cent. 
Even if we assume that the number of income returns for males under 18 was negligible we 
still are faced with a discrepancy difficult to account for. 



OFFICIAL INCOME CENSUSES 399 

concentration of female returns in the lowest income groups which gives 
the flat and modeless appearance to the total curve. The Australian fre- 
quency distribution among males oTily, is much more like our estimated 
American distribution ^ than is the Australian distribution among males 
and females together. Now the concentration of female returns in the 
lower income intervals would seem to be the result of a large number of 
returns made by women and female children receiving petty incomes from 
property who would be classified, in the Australian Census of Population, 
as "dependents" and not as " bread winnei's," ^ 

Of the total female population in 1915, 33.46 per cent made out income 
cards and 23.18 per cent reported positive incomes (10.28 per cent re- 
ported zero or negative incomes). But according to the Austrahan census 
of 1911, only 18.6 per cent of the total female population were classified 
as "breadwinners." Thus the women reporting positive incomes in 1915 
constituted a much larger percentage of the total female population than 
did female "breadwinners" in 1911 of the total female population in that 
year. The discrepancy seems too great to be accounted for by the in- 
crease in the number of women " breadwimiers " caused by the war. More 
than half of the 23.18 per cent of the female population reporting positive 
incomes in 1915 reported incomes under £50 per annum. Moreover, the 
average income of this group was only £22 per annum — under the arith- 
metic average of the interval. This strongly suggests petty incomes from 
property, and part time occupations such as keeping boarders, lodgers, 
chickens, etc., rather than any great increase in the number of female 
"breadwinners." The fact that over 30 per cent of the returns made by 
females reported zero or negative incomes is further evidence that the 
large number of extremely small incomes reported was largely the result 
of the schedule calling for income returns from all persons "possessed of 
property." 

Negative incomes arise in general from business or speculative losses. 
Bad as may be the condition of any laboring class, its members are seldom 
faced with negative incomes. It is unlikely that many of the 249,476 
females reporting "deficit and nil" were wage-earners. They were in 
general the owners of small investments which showed losses, such as 
town lots upon which taxes had been paid.^ 

1 See Income in the United States, Vol. I, pp. 128, 129, 132-135. 

2 All persons are classified as "breadwinners" or as "dependents" by the Australian census. 
Male "breadwinners" in Australia constituted in 1911, according to the census of that year, 
69.4 per cent of the total male population, female "breadwinners" 18.6 per cent of the total 
female population, and total "breadwinners" 45.0 per cent of the total population. These 
figures compare with American census figures for 1910 showing males "gainfully employed" 
to constitute 63.6 per cent of total males, females "gainfully employed" 18.1 per cent of 
total females, and total "gainfully employed" 41.5 per cent of the total population. 

2 It is worth noting that in the Australian schedule "rates and taxes paid" could be de- 
ducted before making an income return. This consideration may be of some importance in 
explaining the very large number of small, zero, and negative incomes. 



400 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

While the frequency curve for Austrahan males is much more like the 
American distribution than the curve representing both male and fem.ale 
Australian income recipients, even it shows a much greater concentration 
in the lowest income intervals than does the American distribution. This 
can probably be accounted for to some extent by a large number of income 
returns for young male "dependents" "possessed of property." 

The essential difference in appearance between the American income- 
distribution curve which we presented in Volume I and the Australian 
curve of 1915 is, then, probably traceable to (1) Australian underreporting 
and (2) Austrahan inclusion of a large number of "dependents" who re- 
ceived petty incomes from property and who were in no important sense 
"breadwinners" or "gainfully employed." 

What shall we say about the desirability or undesirability of including 
in an income frequency distribution dependents receiving petty incomes 
from property? While it is true that their incomes, positive or negative, 
are in a way as real as any other incomes, we must remember that probably 
almost all individuals over six years of age not only receive but earn some 
money income during each year. Shall we then include the entire popu- 
lation over six years old in our distribution? As we approach this theo- 
retical limit it is seen that the concept becomes less and less practically or 
even theoretically interesting. Both practically and theoretically we are 
interested in the incomes of persons who, though they be minors, have 
"economically come of age" and have entered into certain definite rela- 
tions to the machinery of factorial distribution. They are "breadwinners" 
or "persons gainfully employed," and the concept back of such expres- 
sions, though like many economic concepts somewhat of a compromise, 
seems a good compromise for our purposes. 

Defining income recipient as we have, we cannot use the Australian 
material as an aid to the graduation or adjustment of the American income- 
distribution curve in its lower ranges. In the upper income ranges, the 
Australian distribution offers, as we shall see, an interesting illustration 
of the same double swing (letter S) appearance of the curve seen in some 
of the more recent American data.^ 

1 When charted on a double log scale. 



CHAPTER 30 
AMERICAN INCOME TAX RETURNS 

At the beginning of the preceding chapter attention was drawn to some 
reasons wh}^ income-tax returns cannot take the place of an adequate 
income census. Nevertheless tax returns are in many respects the most 
important single source of information we have for estimating the fre- 
quency distribution of incomes. Were there neither tax returns nor in- 
come censuses for any country, it is difficult to see how we could make 
even an interesting guess as to the distribution of income in the upper 
ranges. 

American income-tax data go back to 1913. We have now at our dis- 
posal returns for the seven years, 1913 to 1919, inclusive.^ However, the 
amount of information given in the official reports for the earlier years 
1913, 1914 and 1915 is not great. Little is shown beyond the number 
of returns classified by large income intervals and the same returns classi- 
fied by districts. The 1916 tax report is the most voluminous and in one 
respect the most adequate report which has yet appeared.^ It contains 
a set of tables which we are sorry to miss in the later reports, showing 
the frequency distribution of incomes by separate occupations. Other 
features of this report which have been retained in later years are tables 
showing both number of returns and amount of net income for each income 
class for the country as a whole, and the same by States; tables showing 
the sources of the income returned in each income interval, that is the 
amount from wages, business, property; distribution tables arranged by 
sex and conjugal condition; amounts of tax collected from each income 
class, etc. 

Changes in the Federal Income Tax Law during the period have not 
been such as greatly to affect any conclusions which we have drawn from 
the data. From the standpoint of this investigation, probably the most 
important changes in the law relate to general deductions, professions, and 
minimum taxable income. 

In the 1916 returns all deductions were classified as general deductions. 

^ The Annual Reports of the Commissioner of Internal Revenue are the sources for American 
income-tax data for the years 1913 to 1915. Since 1915 the data have appeared annually 
as a separate Treasury Department publication entitled Statistics of Income. 

2 A peculiarity of the 1916 data is that the returns are tabulated as family rather than in- 
dividual returns. "The net incomes reported on separate returns made by husband and wife 
in 1916 are combined and included as one return in the figures for the several classes." Statis- 
tics of Income, 1917, p. 22. 

401 



402 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

In the 1917 returns the types of deductions classified as general deductions 
were greatly reduced; not even contributions were included. In 1918 the 
category was enlarged; contributions, for example, were again placed in 
the general deductions class. Now these changes affect greatly the rela- 
tions between net and total income from year to year. Reported net income 
was in 1916 only 75.43 per cent of reported total income, in 1917 it was 
92.67 per cent, in 1918 89.74 per cent, and in 1919 88.51 per cent. As 
it is the total and not the net income which in the Statistics of Income, is 
divided up according to source, such fluctuations as the above interfere 
with comparisons of different years. 

While income from professions was tabulated separately in 1916, in 1917 
it was included in wages and salaries, and in 1918 and 1919 in business. 

In the 1913 to 1916 returns exemptions were $3,000 per annum for an 
umnarried person, or a married person not living with his wife (or her 
husband), and $4,000 per annum aggregate exemption for married persons 
living together.^ In the 1917 and later returns these minima were reduced 
to $1,000 and $2,000 respectively. However, the increase in usefulness for 
our purposes of the 1917 and later returns was even greater than the 
lowered minima would suggest. Not only was the minimum taxable 
income lowered from $3,000 to $1,000, but this reduction occurred in the 
face of a rapidly rising general level of incomes. With the rise in incomes, 
$3,000 in 1918 or 1919 was relatively a much smaller income than $3,000 
in 1913. In other words, we might logically expect $3,000 to be relatively 
further down the income distribution curve in 1918 than in 1916 or 
1917. 

The accuracy of the reporting is, of course, a matter of great importance 
for this investigation. Now, while it does not seem possible to measure 
directly from the data changes in accuracy of reporting during the period, 
the rapid expansion of the income-tax organization and its increasing 
attention to the investigation and checking of returns establish the pre- 
sumption of greater statistical value in the reports for the later years. 
Offsetting this to an unknown degree is the apparently increasing amount 
of "legal evasion" in the higher income classes. The reporting for the 
years 1913, 1914, 1915 and 1916 appears to have been pecuharly bad in 
the lower income ranges. The distinct improvement in 1917 (compare 
the 1917 returns with those for earlier years in Tables 28B, 28C, 28D, 28E, 
and Charts 27 and 28 of Volume I) seems associated with the patriotic 
enthusiasm engendered by the war. Upon our entry into the war, not 
only did the Bureau of Internal Revenue make an increased effort to ob- 

1 As the returns for 1913 were for income received for the ten months March 1 to December 
31, 1913, the actual minima used for reporting purposes were $2,500 and $3,333.33 (i. e., \^ 
of $3,000 and $4,000 respectively). 



AMERICAN INCOME TAX RETURNS 403 

tain correct returns but individuals, under the spur of patriotism, seem to 
have made less effort to evade. ^ 

The remainder of this chapter is concerned largely with a discussion 
of possible irregularities in the distribution of non-reporting and under- 
statement in the later years. While the total amount of non-reporting 
and understatement was almost certainly greater in the returns for 1917 
than in those for 1918 and 1919, are we sure that the non-reporting and 
understatement of these later years are not possibly more irregularly dis- 
tributed along the frequency curve than was the case in 1917? Is it 
possible that the improvement in the accuracy of the published returns 
for 1918, as compared with those for 1917, was so much greater in the 
income intervals under $5,000 that the resulting change in the shape of 
the frequency curve may amount to something almost akin to an "over- 
adjustment"? 

Income returns by individuals are made on two types of blanks, a blank 
to be filled iii by persons reporting incomes under $5,000 and another 
blank to be filled in by persons reporting incomes over that figure. Now, 
while the returns of incomes under $5,000 and made on "under $5,000" 
blanks are examined, investigated and audited in the field soon after 
their receipt, the investigation and audit of the returns for incomes over 
$5,000 are handled in Washington. If an individual has an actual income 
of $8,000 but reports $4,600 (on an "under $5,000" blanlv), as soon as a 
Field Collector discovers this discrepancy, he passes the matter over to 
the Revenue Agent in charge of the District for Field Investigation. The 
return, accompanied by the Agent's report, is forwarded to Washington 
for final audit. Thus the Field Collectors audit only returns that are (a) 
made on "under $5,000" blanks and (b) believed, after investigation, to be 
for incomes which are actually under $5,000. 

While the Field Audit of returns of these incomes is well under way 
before the preparation of the statistical tables in the Statistics of Income 
and hence appears in that tabulation to an unknown extent, the Washing- 
ton audit of incomes over $5,000 has hardly begun and hence the amended 
figures for these higher incomes do not appear in the Statistics of Income. 
It is impossible to say exactly how much of the "bulge" - which appears 
in the $1,000 to $5,000 interval on the double log charts of the 1918 and 
1919 tax income distributions is caused by a difference in the accuracy 
of the published figures for returns of incomes under and over $5,000. 
However, the Treasury Department states that "the Statistics of Income 

1 It must not, of course, be assumed that the increase in the number of returns in 1917 is 
traceable solely to increased goodness of reporting. 

2 Described in Chapter 28. At many points in the following discussion the reader should 
refer back to the presentation of the case for heterogeneity in the income-tax data contained 
in Chapter 28. 



404 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

are compiled almost entirely from unaudited returns whether they be for 
'under $5,000' or 'over $5,000.'" It seems probable therefore that the 
sudden change in slope of the 1918 curve (on a double log scale) at about 
$5,000 can be explained only partially by a change in accuracy of the 
published returns at that point. 

Moreover, a considerable amount of evidence, some of which has already 
been presented in Chapter 28, suggests that the "bulge" on the income 
curves for the later years corresponds to a reality on the actual income 
curves. While it may be somew^hat over-accented in the published figures 
for 1918 and 1919, and while the figures for 1917 might have shown more 
of such a "bulge" ^ had the reporting been better, we must not assume 
that the pubHshed figures for either 1917 or 1918 give a radically incorrect 
picture of the facts merely because the income curves for the two j^ears 
are so different. The dogma of the similarity of the income curve from 
year to year has little evidence to support it. 

It is by no means certain that even the apparently definite and sharp 
angles on the curves in this $4,000 to $6,000 region give an unreal picture. 
While it is true that we find the same angles on the wages and salaries 
curve, that curve itself seems heterogeneous. An income distribution 
curve composed of wage and salary earners (in the ordinary sense of the 
terms) may well cut an income distribution curve composed of "salaried 
entrepreneurs," and business and financial experts somewhere in the lower 
income ranges. The angle on the composite curve may give a decidedly 
accurate picture of the facts.- 

Let us see what light the data throw on some of these problems. 
Table 30A showing the number of returns for the lower income intervals 
in 1917, 1918, and 1919 and the percentage movements from year to year 
illustrates the great increase in the number of returns in the under-$5,000 
intervals between 1917 and the later years. 

Chart No. 28 of Volume I, on which are drawn the frequency distributions 
for each year from 1916 to 1919 on a double log scale, shows the difference 
in the appearance of the income curves for the three years. Examining 
that chart we notice that the 1918 data-points, which in the upper income 
ranges run nearly as smoothly as the 1917 points, in the $4,000 to $5,000 
interval move abruptly upwards and from there on into the lowest income 
ranges are well above the 1917 points, showing on the chart an irregular, 
plateau-like effect in these lowest income ranges. No such "plateau" 
is apparent on the 1917 line. The j^ear 1919 presents in that chart a 

1 While the 1917 curve runs much more smoothly in the $3,000 to $6,000 range than either 
the 1918 or 1919 curves, it is not without the hint of a bulge beginning at about $4,500. See 
p. 412. 

- In constructing the complete income distribution curve for 1918, published in Volume I, 
the influence of changes in the accuracy of reporting around $5,000 income was probably 
overestimated. 



AMERICAN INCOME TAX RETURNS 



405 



TABLE 30A 





Number of returns 


Percentage increases 


Income intervals 








1918 


1919 


1919 




1917 


1918 


1919 


over 
1917 


over 
1918 


over 
1917 


$2,000-$3,000 


838,707 


1,496,878 


1,569,741 


78.47 


4.87 


87.16 


3,000- 4,000 


374,958 


610,095 


742,334 


62.71 


21.68 


97.98 


4,000- 5,000 


185,805 


322,241 


438,154 


73.43 


35.97 


135.81 


5,000- 6,000 


105,988 


126,554 


167,005 


19.40 


31.96 


57.57 


6,000- 7,000 


64,010 


79,152 


109,674 


23.66 


38.56 


71.34 


7.000- 8,000 


44,363 


51,381 


73,719 


15.82 


43.48 


66.17 


8,000- 9,000 


31,769 


35,117 


50,486 


10.54 


43.77 


58.92 


9,000-10,000 


24,536 


27,152 


37,967 


10.66 


39.83 


54.74 



similar appearance to 1918 though the absence of small intervals in the 
range immediately above $5,000 disguises the characteristics of the curve 
materially.-^ 

The change in the contour of the lower range of the tax income frequency 
curve from 1917 to 1918 and 1919, is, as we have mentioned, associated 
with a large increase in the relative amount of income from wages and 
salaries in the lower intervals. Tables SOB and 30C are interesting in 
this connection.^ 

The 1916 figures in Table SOB are introduced simply because they 
are computable.^ However, too much weight must not be attached to 
them. The 1916 returns are undoubtedly extremely inadequate. The 
high percentages that year from $S,000 income (the 1916 minimum) up 
to about $10,000 may possibly be the result of the ease with which salary 
returns (as opposed to wage, business, or other returns) are obtainable. 
The $4,000 to $5,000 interval is the lowest comparable interval for the 
four years.^ In that interval the numbers of returns by years were: 

1916- 72,027 
1917-185,805 
1918-S22,241 
1919-438,154 



»When chart 28 was drawn for Volume I, only "preliminary" large interval data were 
available. Final small interval data show a "bulge" very similar to that seen in the 1918 line. 

2 The 1917 official wages figures include income from professions. The 1918 and 1919 wages 
figures do not. This makes the increase in the percentages in 1918 still more striking. In- 
come from professions was tabulated separately in 1916, but was included in the wages figures 
for that year in order that 1916 and 1917 might be comparable. 

3 No data are available from which corresponding figures for 1913, 1914 or 1915 might 
be calculated. 

1 The $3,000-$4,000 interval did not in 1916, include married persons making a joint return. 



406 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



TABLE SOB 



PER CENT THAT INCOME FROM WAGES AND SALARIES IN EACH NEf 
INCOME CLASS WAS OF TOTAL NET INCOME IN THAT CLASS 



Income class 


1916 


1917 


1918 


1919 


$ 1,000-$ 2,000 






79.45 


83.49 


2,000- 3,000 






69.75 


74.53 


3,000- 4,000 


76.98 




55.21 


61.86 


2,000- 4,000 




46.32 


(64.42) 


(69.45) 


4,000- 5,000 


66.88 


36.30 


48.85 


52.48 


5,000- 10,000 


53.31 


36.16 


39.59 


43.24 


10,000- 20,000 


36.38 


32.94 


38.60 


38.11 


20,000- 40,000 


24.60 


26.82 


33.16 


33. 3S 


40,000- 60,000 


17.23 


22 . 74 


27. 8S 


27.57 


60,000- 80,000 


16.20 


19.67 


25.36 


21.01 


80,000- 100,000 


13.37 


18.51 


22.16 


22.70 


100,000- 150,000 


13.34 


15.75 


18.44 


18.75 


150,000- 200,000 


9.39 


12.65 


16.16 


15.42 


200,000- 250,000 


9.14 


12.30 


13.07 


13.62 


250.000- 300,000 


7.87 


9.36 


12 . 57 


11.92 


300,000- 500,000 


6.59 


10.17 


11.27 


10.18 


500,000-1,000,000 


5.21 


6.39 


5 . 42 


6.80 


1,000,000-1,500,000 


4.84 


2.83 


7.51 


1.60 


1,500,000-2,000,000 


3.23 


3.76 


2.21 


10.00 


2,000,000 and over 


.51 


2.39 


.85 


4.02 



The amounts of income from wages and salaries and from other net income 
in the $4,000-15,000 interval were year by year in miUions of dollars: 





1916 


1917 


1918 


1919 


Wages and salaries '^ 


216 
107 


301 

528 


703 
736 


1,029 


Other net income 


931 



Income from professions is included in the 1916 and 1917 wages and salaries figures. 

The percentage changes in these items from one year to the next were: 

1917 1918 1919 

1916 1917 1918 

Wages and salaries 139 . 3 233 . 7 146 . 4 

Other Net Income 493 . 139 . 4 126 . 6 

It is plain that the great increase in the $4,000-$5,000 interval ^ in 1917 
was in income from other sources than wages and salaries. 

Table 30C shov/s the wage and salary figures compared with total income 
instead of net income as in Table 30B. It was, of course, necessary to re- 
tain the net income intervals as the data are not classified in total income 

1 As may be seen from Tables SOB and 30C, the increase from 1916 to 1917 in income from 
other sources than wages and salaries was greater than the increase in income from wages 
and salaries not only in the $4,000-$5,000 interval but also in the $5,000-$10,000 interval. 



AMERICAN INCOME TAX RETURNS 



407 



intervals. Though the relations between years are different in this table 
from what they are in the net income table/ the distribution of the per- 
centages in each individual year shows much the same characteristics in 
both tables. 

TABLE 30C 



PER CENT THAT INCOME FROiM WAGES AND SALARIES IN EACH NET 
INCOME CLASS WAS OF TOTAL INCOME IN THAT CLASS 



Income class 

(Net) 


1916 


1917 


1918 


1919 


$ 1,000- $2,000 






74.67 


77.25 


2,000- 3,000 






65.42 


69.14 


3,000- 4,000 


47.74 




51.14 


56.71 


2,000- 4,000 




41.82 


(60.15) 


(64.12) 


4,000- 5,000 


45.96 


33.60 


44.82 


47.12 


5,000- 10,000 


36.38 


33.87 


33.55 


36.60 


10,000- 20,000 


25.76 


30.89 


33.10 


32.70 


20,000- 40,000 


18.81 


25.20 


28.76 


28.36 


40 000- 60,000 


13.75 


21.23 


23.79 


23.39 


60,000- 80,000 


12.76 


18.56 


21.51 


20.33 


80,000- 100,000 


10.74 


17.61 


19.00 


19.25 


100,000- 150,000 


11.06 


15.05 


15.92 


15.40 


150,000- 200,000 


7.68 


12.01 


13.10 


12.41 


200,000- 250,000 


7.83 


11.75 


11.22 


11.26 


250.000- 300,000 


6.64 


8.71 


10.73 


9.80 


300,000- 500,000 


5.50 


9.59 


9.62 


8.19 


500,000-1,000,000 


4.35 


5. 88 


4.37 


5.38 


1,000,000-1,500,000 


4.12 


2.62 


6.29 


1.34 


1,500,000 2,000.000 


2.82 


3.54 


1.81 


8.54 


2,000,000 and over 


.47 


2.18 


.63 


.32 



The percentages in Tables SOB and 30C show each year a sudden increase 
(as we approach the lower income intervals) somewhere in the $4,000 to 
$5,000 or the $5,000 to $10,000 interval. At exactly what point each year 
do these sudden increases seem to occur? Charts SOD, SOE and SOF pre- 
sent the material in a slightly different form. They illustrate the relation- 
ship between the average income from wages and salaries in each net 
income interval and the average total income in the same net income in- 
terval for the years 1917, 1918 and 1919 on a double log scale. The 1918 
and 1919 charts immediately suggest the improbability of being able to 
describe the data by a single simple mathematical expression. To the 
1918 data-points have been applied two distinct mathematical curves, 
which fit the data remarkably well and intersect at about $6,700 total 
income. The curve fitted to the upper income ranges is a parabola, while 
that fitted to the lower income ranges is an hyperbola, one of whose asymp- 
totes is the 45° line which divides the chart into a "possible" and an "im- 

1 Some reasons for the changes in relation of net to total income from year to year are 
mentioned en pages 401 and 402. 



408 



PERSONAL DISTRIBUTION OF INCOME IN U. S 




AMERICAN INCOME TAX RETURNS 



409 



CHART 30 E 



o 

I- 100 00 
Q 
2 



U.5 INCOME TAX RtTllRNS 
I9IS 

AVERAGE income: 

FROM 

WAGES AMD SALARIES 

AND 

AVERAGE TOTAL inCOME 



EACH NET mCOME INTERVAL 
Scales Logarithmic 




TOTAL INCOME IN THOUSANDS OF DOLLARS 
20 30 4U 50 100 200 300 40U 500 



1,000 2,O0U 3.0004,000 



410 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



CHART 30F 



-40S 



U.5 INCOME TAX RETURNS 
1919 
AVERAGE mCOME 

FROM 

WAGES AMD SALARIES 
AVERAGE TOTAL INCOME 

IN 

EACH MET mCOME INTERVAL 
Scales Logarithmic 




TOTAL INCOME IN THOUSANDS OF DOLLARS 
20 30 40 50 100 200 300 400 300 



1.000 2,000 3,000 4,000 



AMERICAN INCOME TAX RETURNS 



411 



possible" area. The equations of the two (1918) curves on a double log 
scale are (I) y + 3.92945 — 2.744 x-{- .22x' =0 (parabola) 

(II) ij - — 3.981909 y — .867246 xy + 3.981909 x — .132754 x ^ 
— .060262 = (hyperbola) 
As it is difficult to estimate accurately by eye the goodness of fit of a curve 
to data when charted on a log scale, Table 30E is introduced : 

TABLE 30E 

WAGES AND INCOME IN THE 1918 INCOME TAX RETURNS 







Average income from 








wages and salaries 


Percentages 


Not income 


Average 
total income 






tnaf rifit^ irp of 


intervals (1918) 






mathematical 






Data 


Mathematical 
curves 


curves 


$ 1,000-$ 2,000... 


$ 1,566 


$ 1,169 


$ 1,178 


99.2 


2,000- 3,000 . . . 


2,583 


1,690 


1,652 


102.3 


3,000- 4,000 . . . 


3,710 


1,897 


1,955 


97.0 


4,000- 5,000... 


4,866 


2,181 


2,117 


103.0 


- 5,000- 6,000... 


6,388 


2,192 


2,216 


98.9 


6,000- 7,000... 


7,620 


2,537 


2,555 


99.3 


7,000- 8,000... 


8,952 


2,963 


3,012 


98.4 


8,000- 9,000... 


10,148 


3,341 


3,407 


98.1 


9,000- 10,000... 


11,214 


3,747 


3,760 


99.7 


10,000- 11,000... 


12.207 


4,171 


4,078 


102.3 


11,000- 12,000... 


13,707 


4,555 


4,542 


100.3 


12,000- 13.000 . . . 


14,263 


4,806 . 


4,709 


102.1 


13,000- 14,000 . . . 


15,922 


5.529 


5,204 


106.2 


14,000- 15,000... 


16,778 


5,801 


5,455 


108.3 


15.000- 20,000... 


20,167 


6,-375 


6,400 


99.6 


20,000- 25,000... 


25,859 


7,891 


7,860 


100.4 


25,000- 30,000... 


31,704 


9,196 


9,211 


99.8 


30,000- 40,000... 


39,644 


10,711 


10,872 


98.5 


40,000- 50,000... 


52,319 


12,639 


13.192 


95.8 


50,000- 60,000 . . . 


64,327 


14,963 


15,066 


99.3 


60,000- 70,000 . . . 


74,848 


16,576 


16,539 


100.2 


70,000- 80,000... 


90,4.37 


18,764 


18,459 


101.7 


80,000- 90,000 . . . 


98,379 


19,273 


19,.351 


99.6 


90,000- 100,000... 


111,515 


20,447 


20,682 


98.9 


100,000- 150,000... 


139,520 


22,212 


23,163 


95.9 


150,000- 200,000... 


211,959 


27,758 


27,829 


99.7 


200,000- 250,000 . . . 


259,487 


29,107 


30,088 


96.8 


250,000- 300,000... 


317,578 


34,076 


32.226 


105.7 


300,000- 400,000. .. 


409,756 


44,393 


34,786 


127.6 


400,000- 500,000 . . . 


514,882 


38,967 


36,847 


105.8 


500,000- 750,000... 


765,905 


27,.582 


39,765 


69.4 


750,000-1,000,000... 


1,013,846 


61,183 


41,229 


148.4 


1,000,000-1,500,000... 


1,426,182 


89,710 


42,199 


212.6 


1,500,000-2,000,000. . . 


2,084,715 


37,118 


42,199 


88.0 


2,000 000-3,000,000... 


3,263,673 


50,178 


40,729 


123.2 


3,000,000-4,000,000. .. 


4,515,732 


11,013 


38,753 


28.4 



The data of table 30E move rather erratically in the intervals above 
$300,000 per annum income. This is natural in view of the small number 



412 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



of cases in these upper intervals. There were only 627 returns reporting 
net incomes of over $300,000 per annum; this is less than one seventieth 
of one per cent, of the total number of returns. In the 28 intervals under 
1300,000 per annum 14 of the percentages show the data within one and 
one half per cent, of the mathematical values. 

These mathematical curves have not been introduced as being in any 
sense the "law" of the data but merely to emphasize how smoothly the 
data curves run and yet how unmistakable a sensation they give us of two 
parts, one above about $6,700 total income and one below that figure.-^ 
It would, of course, be quite impossible to get any sort of approximation 
to the lower range data by producing the parabola fitted to the upper 
income ranges. How impossible may be seen from Table 30EE. 

TABLE 30EE 

WAGES AND INCOME IN THE 1918 INCOME TAX RETURNS 



Net income 
intervals (1918) 



$4,000-15,000 
3,000- 4,000 
2,000- 3,000 
1,000- 2,000 



Average 
total 



$4,866 
3,710 
2,583 
1,566 



Average income from wages 
and salaries 



Data 



^2,181 
1,897 
1,690 
1,169 



Hyper- 
bola 



P,117 
1,955 
1,652 
1,178 



Para- 
bola 



51,574 

1,152 

745 

391 



Percentages that 
data are of 



Hyper- 
bola 



103.0 
97.0 

102.3 
99.2 



Para- 
bola 



138.6 
164.7 
226.8 
299.0 



The 1919 data show the same two-curve appearance as the 1918 data. 
This may be clearly seen from chart 30F.^ The intersection of the two 
curves would be at about $7,100 instead of $6,700 as on the 1918 chart. 
Is there any sign of such a change from one curve to another on the 1917 
data? There seems to be. Chart SOD shows the 1917 data with a parabola 
fitted to the observations above the first interval. This curve and Table 
30D give us a strong impression that the first interval cannot be described 
by any simple curve which describes the remainder of the data. The same 
two-curve characteristics as the 1918 and 1919 data are strongly suggested. 

The equation of the 1917 parabola on a double log scale is t/ + 1.8417 — 
1.8346 X + .124 x^ = 0. The poorness of the fit to the first interval and 
the comparative goodness of the fit to the remainder of the data as high as 
$250,000 per annum may be seen from Table 30D. If the data were 
numerous enough to permit us fitting two curves they would probably 
intersect at about $4,500. 

» An alteration in the size of the intervals in which the data are quoted by the Income Tax 
Bureau would of course change the data curve to some extent. However, taking the intervals 
as they come and fitting the curves to them we get the unmistakable impression of great regu- 
larity. It seemed scarcely worth while to fit the curves to areas rather than points. 

2 The story told by Chart 30F is so plain it seemed hardly necessary to fit another set of 
curves. 



AMERICAN INCOME TAX RETURNS 



413 



TABLE SOD 



WAGES AND INCOME IN THE 1917 INCOME TAX RETURNS 



Net income 


Average 
total income 


Average income from 
wages and salaries 


Percentages 
that data are of 


intervals (1917) 


Data 


Mathematical 
curve 


mathematical 
curve 


$ 2,000-$ 4,000... 

4,000- 5,000... 

5,000- 10,000... 

10,000- 20,000... 

20,000- 40,000... 

40,000- 60,000... 

60.000- 80,000... 

80,000- 100,000... 

100,000- 150,000... 

150,000- 200,000... 

200.000- 250,000. .. 

250;000- 300,000... 

300,000- 500,000 . . . 

500,000-1,000,000. .. 

1,000,000-1,500,000... 

1,500,000-2,000,000... 

2,000,000 and over. . . . 


$ 3,059 

4,818 

7,210 

14,623 

29,236 

51,940 

72,811 

93,742 

126,979 

181,156 

233,880 

293,905 

398,517 

740,769 

1.294,619 

1^812,388 

4,551,718 


$1,280 

1,619 

2,442 

4,517 

7,368 

11,024 

13,516 

16,510 

19,108 

21,758 

27,501 

25,587 

38,204 

43,558 

33,973 

64,201 

99,132 


$1,101 

1,688 

2,422 

4,374 

7,411 

11,038 

13,699 

15,992 

19,081 

23,147 

26,388 

29,478 

33,877 

43,632 

52,845 

58,358 

71,945 


116.3 

95.9 

100.8 

103.3 

99.4 

99.9 

98.7 

103.2 

100.1 

94.0 

104.2 

86.8 

112.8 

99.8 

64.3 

110.0 

137.8 



Both the regularity of the data curves and the positions of the inter- 
sections of the mathematical curves ^ might suggest that heterogeneity 
of the wages and salaries data was the primary cause of the irregularity 
in the total income curve. The position of the points of intersection of the 
mathematical curves might seem inconsistent with a sudden change in 
accuracy of reporting at exactly $5,000. 

However this argument does not appear so conclusive when we examine 
the actual amount of wages in each income interval. The constitution of 
the reported income each year may be seen rather plainly in Charts 28T, 
28U, 28V, 28W, 28X, 28Y, 28Z, and 28AA.2 These charts show the number 
of dollars per dollar income interval reported in each income interval by 
sources for the years 1916 to 1919.^ They not only illustrate the fact that 
the constitution of the income curve changes radically as we move from 
small to large incomes but also picture the salient characteristics of these 
changes; each source curve, being charted on a double log scale, may be 

•' Particularly the 1919 intersection which is above the $5,000 to $6,000 net income interval. 

2 See pages 385 to 392. 

3 The five lines representing wages, business, rents, interest, and dividends were found to 
interweave to such an extent when drawn on one chart that two charts were drawn for each 
year, one representing wages and business and the other incomes from property. 

Wages includes "salaries, wages and commissions" and in 1916 and 1917 "professions and 
vocations." 

Business includes "business," "partnerships, personal service corporations, estates, and 
trusts," and "profits from sales of real estate, stocks, bonds, etc.," and in 1918 and 1919 
"professions." 

Rents includes royalties. 

Interest includes unclassified investment income. 



414 PERSONAL DISTRIBUTION OF INCOME IN U. S. 

seen at a glance in its entirety. We see from Charts 28X and 28Z that, 
though the ratio of the income from wages and salaries to total income 
may, when charted, show an angle above $5,000, the entire "bulge" on 
the wages and salaries curve itself occurs in the under-$5,000 intervals 
both in 1918 and 1919. Moreover, while ''wages and salaries" is the larg- 
est item in these lowest income intervals, and hence is the controlling factor 
in determining the peculiar shape of the total curve in this region, it is not 
the only item showing irregularities and "bulges." Some of these move- 
ments are extremely difficult to explain. Why should a "bulge" appear 
on the lower income ranges of the "rent" curve in 1918 and by 1919 be- 
come pronounced? ^ The appearance of a bulge on the ivage curves in 
1918 and 1919 seems quite explicable on the basis of heterogeneity within 
the wage and salary data themselves but one feels a shade less confidence 
in any explanation of why that curve moved in this peculiar manner if the 
explanation does not seem also clearly applicable to the rents curve which 
moved in an apparently similar m.anner. 

' A mere increase in rents will not, of course, account for this unevenness in their distribu- 
tion. 



CHAPTER 31 

INCOME DISTRIBUTIONS FROM OTHER SOURCES THAN 
INCOME TAX RETURNS 

Concerning the frequency distribution of incomes over $3,000 or $4,000 
per annum we have almost no information aside from the income tax 
returns. Existing wage distributions and non-tax income distributions 
almost never reach higher than $2,500 or $3,000 per annum. 

Even in the lower income ranges (under say $2,500 or $3,000) most of 
the existing non-tax income distributions are of little use in our problem. 
In the first place there are less than half a dozen distributions of this sort 
which are not such small samples as to prevent us feeling much confidence 
in their representative nature.-^ An even more serious defect of every such 
distribution known to us, with one exception^ is that the purpose for which 
the data have been collected almost inevitably makes them extremely 
ill-adapted to our use. For example, one of the largest recent samples is 
prefaced by almost a page of introduction explaining what types of re- 
cipients were purposely excluded.^ This is rather typical. To base upon 
such distributions any wide generalizations with respect to the income 
curve for the country as a whole or even for the localities from which such 
data were collected would be unwarranted. 

Furthermore, almost without exception these studies in income distri- 
bution are on a jamily basis. While it is sometimes possible to make a 

1 For example, Chapin's well-known investigation into the distribution of incomes includes 
only 391 worldngmen's families, and the best distribution of farmers' incomes includes only 
401 farmers from a single state. 

2 Arthur T. Emery's distribution of income among 1960 Chicago households. 

3 "In studying the sources of income and the importance of each source with relation to 
the total income of a family the following limitations to the type of family schedules should 
be kept in mind. No families were scheduled in which there were children who lived as 
boarders, that is, paid a certain sum per week or per month for board and spent the remainder 
of their earnings or salary as they saw fit. No families were scheduled which kept any board- 
ers. The number of lodgers to be kept by a family was limited to three at any one time. No 
families were scheduled in which the total earnings of the family did not equal 75 per cent, or 
more of the total income. It will be seen that these limitations excluded a large number of 
families and this materially affects the percentage of families having earnings from children 
and income from lodgers, and also results in showing a larger percentage of the total income 
as coming from the earnings of the husband than would be the case if the type of families 
named had not been excluded from the study. It also reduces the actual amount per family 
earned by children and received from boarders or lodgers that would be shown in case a cross 
section of a community including all the types mentioned were used. The object in making 
the exclusions named was to secure families dependent for support, as largely as possible, 
upon the earnings of the husband. Of course, it was impracticable to secure a sufficient 
number of families in which the only source of income was the earnings of the husband, but 
in following the course named the percentage of families having an income from other sources 
has been very largely reduced." "Cost of Living in the United States — Family Incomes," 
Monthly Labor Review, Dec, 1919, p. 30. 

415 



416 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



rough estimate of the individual incomes from the family data, such es- 
timates are not what are needed for our purposes. They can show nothing 
but the distribution of income among the individuals constituting these 
families and these families are almost inevitably so chosen as to make the 
individuals composing them not representative of income recipients at 
large. Analysis of the distribution of earnings among the individual mem- 
bers of such families discloses an heterogeneity so extreme as to result in a 
pronouncedly duomodal distribution curve. The fathers' incomes have one 
mode while the children's incomes have another. Chart 31A showing a 
natural scale frequency distribution of earnings among 2811 individuals 
in 2170 families in 1918 ^ exhibits this duomodal appearance in a striking 
manner. The "families" had been so chosen as to exclude both young 



FREttOENCY DISTRIBOTfON 

OF 

AHNUAL EARMIIiGS OF 28)1 IHDIVJDUALS 

IM 

2170 FAMILIES IM THE U.S. IN 1918. 

iOunCE: BuREPU orlflBORST»TISTIC% 

ScfiLEi ffnrcjRni. 



\ 




600 



ANNUAL EARNINGS 

1,200 1,400 




married couples having no children and unmarried but independent wage 
earners. Investigations planned to bring out the economic character- 
istics of such "typical families," while they may be extremely valuable 
for the purposes for which they were undertaken, are necessarily of but 
little use in the construction of a frequency distribution of all individual 
incomes in the community. Moreover, even if we were attempting to 
construct a family and not an individual distribution these data would not 
generally be particularly helpful for, in addition to the exclusions just 
mentioned, further narrow and rigid restrictions are usually, and for the 
purposes in view quite properly, imposed upon the definition of the "typical 
family." 

^ This is a sample from the 12,096 white families referred to in note 3, page 415 The 
detailed figures of this sample were tabulated for us by the Bureau of Labor Statistics. 
They cover 15 cities chosen as representative of the whole list. Each one of the 15 cities 
shows the duomodal appearance referred to in the text. 



DATA FROM OTHER SOURCES THAN TAX RETURNS 417 

As incidentally remarked above, there is one non-tax income frequency 
distribution to which many of the above criticisms do not apply. It is 
the distribution of income among 1960 Chicago ''households" in 1918 from 
an investigation made by Mr. Arthur T. Emery for the Chicago Daily 
News} Instead of attempting to describe a ''typical family" Mr. Emery 
attempted to discover the "household" income of each person whose name 
came at the top of a page in the Chicago city directory. Mr. Emery en- 
countered many difficulties in attempting to follow out this scheme and 
has himself pointed out sources of error.^ Notwithstanding the inevitable 
difficulties, Mr. Emery seems to have made a real effort to obtain a scien- 
tific sample. While his distribution shows unmistakable irregularities, 
it is in many respects for our purposes the most interesting and suggestive 
recent non-tax income distribution available. 

Finally, it seems impossible to obtain from these distributions any but 
extremely general conclusions concerning the relation between income 
from effort and income from property. The data have almost always ^ 
been so chosen as to eliminate any families obtaining an appreciable frac- 
tion of their income from property. While they may give us some clues 
as to the shape of the upper range tail of the wage-earners' income distri- 
bution curve ^ they can tell us little about even the upper tail of the general 
income curve and almost nothing about the lower income tail of either the 
wage-earners' or the general income curve. 

^ While the Bureau is not at liberty to publish this material we were permitted to make 
what use we could of it in constructing our income curve for the country. 

2 In a letter to the Bureau he writes, "There was, however, one important source of error 
in this method— the poorer and middle class residents were willing to talk, and with the care- 
fully trained approach of the investigator, the upper class was also won over, but we found 
in the wealthy districts that the butler and "not at home' caused a large amount of travel 
on the part of the investigator," and often a final failure to obtain any report. 

3 These remarks do not apply to the distribution of income among the 401 farmers or Mr. 
Emery's distribution. However, the Bureau has no figures, in the case of Mr. Emery's dis- 
tribution, for income from property. 

« Compare pages 378, 379, 380. 



CHAPTER 32 
WAGE DISTRIBUTIONS 

There is in all an immense amount of American wage data. On the other 
hand, as an investigator gets into his subject, he begins to realize that the 
material is more remarkable for its fragmentary nature than for its amount 
— great as that may be. For no recent year can he obtain wage distribu- 
tions for more than about 8 per cent, of those gainfully employed. Of 
course, if these 8 per cent, were scattered over the different types of em- 
ployment and localities in any truly random fashion, and if their wages 
were uniformly reported, much might be done with the material. As 
things are, however, whole occupations as important as agricultural labor 
and trade are almost unrepresented. Moreover, as we are interested in 
the amount of wages actually received during the year, it is rather dis- 
couraging to find that this is the one type of distribution which practically 
never occurs. Distributions of amounts actually earned in a month are 
almost as rare. There are a few distributions of amounts actually earned 
in a week or fortnight, but the great majority of wage distributions are 
distributions of wage rates — figures by the how^ being the commonest — or 
of hypothetical earnings, generally known as full-time earnings per week. 

Now it is in general impossible to construct a wage distribution for earn- 
ings from a distribution of rates. Earnings depend, of course, not only on 
rates but also on hours worked. However, we seldom know anything about 
the distribution of hours worked and almost never do we know anything 
about the relation between rates and hours worked. Chart 32A illustrates 
how violent m^ay be the difference in shape of the earnings and rates curves 
for the same individuals.^ The earnings distribution in this particular 
case shows not only a much greater scatter than the rates distribution but 
is of an entirely different shape, as may be seen from Chart 32B where the 
data are drawn on a double log scale. Chart 32C shows the distribution 
of hours worked in a week for the same individuals. Now, though the 
slaughtering and meat packing industry may be an extreme example, 
what evidence we have suggests that distributions of rates and of earnings 
are rarely in close agreement. Moreover the relation of the one distribu- 
tion to the other changes as we pass from industry to industry.^ 

1 43,063 Male Employees in the Slaughtering and Meat Packing Industry in 1917. Bureau 
of Labor Statistics, Bulletin 252. For purposes of comparison the two distributions are so 
placed that the frequency curves show the same arithmetic means and areas. 

2 Resulting largely, of course, from the varying types of distributions of hours-worked-in- 

418 



WAGE DISTRIBUTIONS 



419 



CHART 32 A 

FREQUENCY DISTRIBUTIONS OF RATES OF 
WASES PER HOUR AND EARNINGS PER WEEK 
FOR 4.3,063 MALE EMPLOYEES IN THE 
SLAUGHTERING AMD MEAT PACKING INDUSTRY 
IN THE U.S. IM 1917. 

SqUKCE: BuRBDU Of IflBOIf St/TTUT/CS. 3cnL£TW ZS2 






\ 



\. 



\ 



■"■"I. 



CHART 321 

FREQUENCY DISTRIBUTIONS OFRATES OFWAGES PER HOUR 

AMD EARNINGS PER WEEK FOR 43,063 MALE EMPLOYEES 

IM THE SLAUGHTERING AND MEAT PACKING INDUSTRY IN 

THE U.S. IN 1917. 

Scarce- BuKEnu of L/ibor Sr/jT/sncs Bulletin zsz 

SctiJ.£S LoGHi?irHMic. 



i I.. 




420 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



-5,000 



4,000 2 

s 



FREQUEMCV DISTRIBUTIOM 

OF 

HOURS WORKED IN A WEEK 

FOR 

43,063 MALE EMPLOYEES 

IN THE 

SLAUGHTERING AND MEAT PACKING INDUSTRY 

INTHE 

U.S. m 1317. 

Source :8ui?£Fiu of I/jbo/? Sr/rr/sr/cs 

Bui.LETIN,2SZ 

■Sculps VaruKfiu 



3,000 



■2,000 z 



-1,000 




HOURS WORKED IN THE WEEK 
40 50 60 



The same difficulty as we find in any attempt to estimate the distribu- 
tion of earnings per week from the distribution of rates per hour seems 
inherent in any attempt to estimate the distribution of earnings in a year 
from the distribution of earnings in a week. The unknown distribution of 
weeks worked in the year must seriously affect our results.-^ 

Estimating the frequency distribution of wages earned in a year for an 
industry from the frequency distribution of wages earned in another year 
in the same industry, if we had such data, would involve us in a similar 
difficulty. Even though we knew the total number of individuals gainfully 
employed and their total wage bill each year and also the frequency dis- 
tribution of earnings for one of the years, estimating the frequency dis- 
tribution for the other year would be hazardous. While some rates dis- 
tributions for the same industry in the same locality show symptoms of 
not changing in shape very radically from year to year,^ this does not seem 



the-week (month or year) in different industries. Illustrations of lack of uniformity in the 
relation between rates and earnings of the same persons for the same period but in different 
industries were worked up from Professor Davis R. Dewey's Special Report on Employees 
and Wages for the 12th Census. 

^ We have no distributions of amounts earned in a week and in a year for the same industry, 
with which to illustrate this point directly. 

2 For example, the distribution curve for wages per week among Massachusetts factory 
workers shows a moderate degree of similarity of shape from year to year. 

Professor H. L. Moore (Political Science Quarterly, vol. XXII, pp. 61-73) discussed the 
fluctuation from 1890 to 1900 in the variability of wage rates in a total made up of thirty 



WAGE DISTRIBUTIONS 



421 



a sufficient reason for assuming the same of earnings distributions. The 
shape of the distribution representing hours or days worked in the year 
may be expected to change greatly from year to year with alternations of 
prosperity and depression.^ 

What little evidence we possess suggests that wage distributions ^ for 
individuals of the same sex in the same industry at the same date, but in 
different localities, though generally more dissimilar in shape than distri- 
butions for the same industry in the same place but at different dates, 
are less unlike one another than distributions for different industries though 
in the same place and at the same time. The variation in shape of such 
distributions for different industries is often extreme.^ 

selected manufacturing industries. These distributions (for 1890 and 1900) illustrate both 
the similarity and the difference in rates distributions between the two years. 

^ For example, what little information we have points to the "scatter" of the days-worked- 
in-a-year distribution being much greater in a year of depression than in a year of prosperity. 

The extreme variations in shape of the income distributions for the same 1240 individuals 
in the years 1914 to 1919 as seen in the Statistics of Income, 1919, page 30, are interesting in 
this connection. 

2 Whether earnings or rates. 

' Examples of this are numerous. Charts 32D and 32E show the distribution of wages 
per week among Massachusetts males working in (a) the boot and shoe industry and (b) the 
paper and wood pulp industry. For purposes of comparison the two distributions are so 
placed on the natural scale chart that the frequency curves show the same arithmetic means 
and areas. The double log chart is based directly upon the natural scale chart. It was 
necessary to break up the "over $35" interval before calculating the arithmetic means. 



FREQUEMCY DISTRIBUTIOM 

OF 

RATES OF WAGES PER WEEK 

FOR 

MALES IN THE BOOT AND SHOE INDUSTRY 
AND TOR 
MALES IN THE PAPER AMD WOOD POLP INDUSTRY 
IN MASSACHUSETTS IN I9I& 

So<IIK£: MffSSIKTIC/lCTTS STHT/STIC3 Or M/tHVmCmXCS WIB 

Scales ArnruKHt. 




422 



PERSONAL DISTRIBUTION OF INCOME IN U. S. 



CHART 32 E 
FREQUENCY DISTRIBUTION OF RATES OF WAGES PER WEEK 
FOR MALES IN THE BOOT AND SHOE INDUSTRY AND FOR MALES 
IN THE PAPER AMD WOOD PULP INDUSTRY IN MASSACHUSETTS 

IN I9IS. 

Source.. M/T'5SffcHus£TTs Strtist/cs of M/JNaF0CTa/?£S /9/e. 




In conclusion, the order of importance of the variables as affecting the 
shape of the distribution curve seems to be — industry, place, time. 

We have but little basis for estimating total income from earnings. In 
the preceding chapter on Income Distributions from other Sources than 
Income Tax Returns attention was drawn to the difficulty of arriving at 
any reliable statement of relationship between earnings and income from 
such distributions because of the way in which the data were selected. It 
is even less possible to discover the nature of any such relationship from the 
income-tax material. Though there is no such apparent "selection" in 
the income-tax data as in the case of non-tax income distributions, the 
material is not arranged to answer our particular question. 

The non-statistical reader examining Charts SOD, 30E and 30F, on which 
are plotted average total income and average income from wages in each 
income interval, might think that it would be quite simple to estimate the 
probable average total income of persons having any specified wage. How- 
ever there is a profound statistical fallacy involved in the use of this ma- 
terial for any such purpose. As given in the official tables, income is the 
independent variable, wages the dependent. This condition cannot be 
reversed without retabulation of the original returns. The statistical 
student recognizes the problem as one involving the impossibility of de- 
riving one regression line from the other when neither the nature of the 



WAGE DISTRIBUTIONS 423 

equation representing the regression line ^ nor the degree of relationship 
(correlation in the broad, non-linear sense) is known. Even if we knew 
that the average net income of those persons reporting in 1918 in the $5,000 
to S6,000 net income class was $5,474 and the average wage obtained by 
these persons was $2,192, we would be quite unwarranted in concluding 
that the average income of persons receiving $2,192 per annum loages was 
$5,474. If no wage earner received income from any other source than 
wages we still would have a condition where the average income in the 
income class would be greater than the average wage. Total wages would 
be necessarily less than total income, because in the income class are in- 
cluded not only wage earners but capitalists and entrepreneurs. But both 
total wages and total income are divided by the same number to get an 
average — namely total number of persons in that income class. 

This suggests a technical criticism of the material contained in the 
Statistics of Income. All data concerning the relation between two vari- 
ables are always there published in such a manner as to give information 
concerning only one of the regression lines and no information whatever 
concerning the "scatter." If such data were published in the form of " cor- 
relation tables" the increase in usefulness for statistical analysis would 
be very great. Such "correlation tables'-' keep closer to the original data 
than the usual type of statistical tables. Freer use of them is much to be 
desired, particularly in cases where it is difficult to anticipate all the prob- 
lems for whose solutions investigators will go to the tabulated materials. 

1 The difficulty of the problem is, if possible, increased in this particular case because of 
the fact that the regression is radically non-linear. 



CHAPTER 33 

THE CONSTRUCTION OF A FREQUENCY CURVE FOR ALL 
INCOME RECIPIENTS 

The direct and only adequate method of discovering what is the fre- 
quency distribution of income in the United States would be to define 
very carefully the terms income and income recipient and then have a care- 
fully planned census taken by expert enumerators upon the basis of these 
definitions. The returns brought in by the enumerators should moreover 
be sworn to by the persons making them and heavy penalties attached to 
the making of false or inaccurate returns. A less satisfactory method but 
one which would probably give excellent results would be to have a large 
number of truly random samples taken by such a census. The results of 
either procedure could then be adjusted in the light of other statistical 
information concerning the National Income and also in the light of theo- 
retical conclusions derived from the data themselves. 

Constructing an income frequency distribution for all income recipients 
in the United States from the existing data, a few of whose peculiarities 
have been noted in the preceding chapters, necessarily involves an ex- 
tremely large amount of pure guessing. It is only because of the practical 
value of even the roughest kind of an estimate that any statistician would 
think of attacking the problem. The method followed in the actual con- 
struction of the income frequency distribution has been outlined in vol- 
ume I.^ This method contains one assumption after another that is open 
to question. Moreover we feel in many cases quite unable to estimate* 
the probable errors involved in these assumptions. Their only excuse is 
their necessity. What is the amount of under-reporting for income tax 
and how is it distributed? What is the effect upon the returns of "legal 
evasion?" To what extent is the "bulge" on the income-tax returns in the 
region under about $5,000 in 1918 the result of the "intensive drive?" 
What is the relation between wages and total income by wage intervals? 
What is the relation between wage rates and earnings in any particular 
industry? Etc., etc. These are all questions which must be answered over 
and over again and yet they are questions the answers to which must be, 
in many instances, almost pure guesses. And, to repeat, the margin of 
possible error is often large. 

In view of the sparsity and inadequacy of the data, our first approach 
to the problem was an attempt to discover, if possible, some general mathe- 
matical law for the distribution of income. Were we to get any very defin- 

1 Income in the United States, Vol. I, pp. 122-139. 

424 



FREQUENCY CURVE FOR INCOME RECIPIENTS 425 

ite and reliable clues as to the mathematical nature of the frequency dis- 
tribution of income from small sample income distributions and from wages 
distributions, etc., such clues might of course be invaluable in checking 
the results obtained from piecing together existing wage distributions, 
income distributions, and other scattered information. We would be in 
the position of the astronomer who is able to "adjust" the results of his 
observations in the light of some known mathematical law. It soon be- 
came clear, however, that it is quite impossible to discover any essential 
peculiarities of the income frequency distribution. The available material 
is not only insufficient for purposes of such generalizations, but moreover 
the distribution from year to year is so dissimilar, that any generalization 
of this nature is too vague to be of any practical value. 

The method finally used for the construction of the income curve has 
therefore, we are sorry to say, practically all the weaknesses of the data 
from which it has been constructed. The occupations of the country 
were tabulated and to each occupation was assigned those wage and income 
distributions which seemed applicable with the least strain. We had then 
a series of income and wage distributions which nominally covered nearly 
all the income recipients in the United States, though for some occupations 
the inadequacy of the wage and income samples was little short of absurd. 
The wage distributions were converted into income distributions on the 
assumption that the smaller the wage the larger is its percentage of total 
income. Beyond this simple assumption the particular functional relation- 
ships used for many industries were almost pure guess work. Moreover, 
not only was there the danger of error in moving from wage distribution 
to income distribution and the danger of error resulting from estimating 
a wage distribution for a particular industry in a particular locality from 
a similar though not identical industry in a different locality, but also there 
was the danger of error resulting from estimating a wage distribution for 
one year from a wage distribution for another. 

The final results are probably not quite so bad as they might have been 
had we not had a number of collateral estimates with which roughly to 
check up and otherwise adjust the first results of our estimates. For ex- 
ample, such independent information as Mr. King's estimate of the total 
income of the country and Mr. Knauth's estimate of the total amount of 
income from dividends were pieces of information with which the results 
of the frequency curve calculations were made to agree. 

Some hypothetical reasoning is inevitable in such a statistical study as 
the present one. The investigator must not lose heart. Sir Thomas 
Browne in his rolling periods sagely remarks that "what song the Syrens 
sang, or what name Achilles assumed when he hid himself among women, 
though puzzling questions, are not beyond all conjecture!" 



VITA 

Frederick Robertson Macaulay was born in Montreal, Canada, August 12, 1882. He 
attended McGill University, 1899-1902; Colorado College, 1906-1907; the University 
of Arizona, 1907-1908; the University of Colorado, 1908-1911. From the University 
of Colorado he received three degrees, B. A. 1909, M. A. 1910, LL. B. 1911. 

He attended Columbia University for three years, 1912-1915. During that time he 
studied under Professors Edwin R. A. Seligman, Benjamin M. Anderson, Jr., Robert E. 
Chaddock, John B. Clark, William A. Dunning, Frank A. Fetter, Franklin H. Giddings, 
Wesley C. Mitchell, Henry L. Moore, Henry R. Mussey, Karl F. Th. Rathgen, James H. 
Robinson, Joseph Schumpeter, Henry R. Seager, James T. Shotwell, Vladimir G. 
Simkhovitch. He attended the seminars of Professors Seligman, Seager, and Schum- 
peter. 

He taught miscellaneous economic subjects for one year (1915-1916) in the University 
cf Washington, Seattle, Washington. He then taught Economic Theory and Statistics 
for three years (1916-1917, 1917-1918, and 1919-1920) in the University of California, 
Berkeley, California. During the year 1918-1919 he was California District Statistician 
for the Emergency- Fleet Corporation. Since Maj', 1920, he has been on the research 
staff of the National Bureau of Economic Research, New York City. 



LIBRARY OF CONGRESS 



013 779 176 6 



